Quadratic Equation
- What is a Quadratic Equation?
- The `3` Forms of Quadratic Equations
- What are the Roots of Quadratic Equations?
- The Quadratic Formula with Explanation
- Deriving Quadratic Formula
- The Discriminant
- Sum and Product of the Roots
- Writing Quadratic Equations Using Roots
- Different Methods to Solve Quadratic Equations
- Factoring Method
- Completing the Square Method
- Solve Quadratic Equations by Graphing
- Square Root Method
- Practice Problems
- Frequently Asked Questions
What is a Quadratic Equation?
A quadratic equation is a type of algebraic equation that involves a variable raised to the second power, often written in the standard form as \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants and \( x \) is the variable.
Quadratic equations are fundamental in algebra and are used to solve problems involving areas, velocities, forces, and many other real-world applications. They often represent the path of a projectile, the shape of a parabola, or the motion of objects under the influence of gravity.
The `3` Forms of Quadratic Equations
There are three commonly used forms of quadratics:
1. Standard Form:
The standard form of a quadratic equation is:
\(y = ax^2 + bx + c = 0 \)
where \(a\), \(b\), and \(c\) are constants, and \(a \neq 0\).
2. Vertex Form:
The vertex form of a quadratic equation is given by:
\( y = a(x - h)^2 + k \)
where \((h, k)\) represents the vertex of the parabola.
3. Factored Form:
The standard form of a quadratic equation is:
\(y = a(x - r_1)(x - r_2)\)
where \(r_1\) and \(r_2\) are roots of the equation, and \(a \neq 0\).
What are the Roots of Quadratic Equations?
The roots of a quadratic equation are the values of the variable \( x \) that satisfy the equation, meaning they make the equation true when substituted into it. These roots represent the points where the graph of the quadratic function intersects the `x`-axis.
Explanation: Consider the quadratic equation in the standard form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. To find the roots of this equation, one typically employs methods such as factoring, completing the square, or using the quadratic formula.
Factoring: If the quadratic expression can be factored into two linear factors, then setting each factor equal to zero gives the roots of the equation.
Completing the Square: This method involves transforming the quadratic equation into a perfect square trinomial, from which the roots can be directly determined by taking the square root.
The Quadratic Formula with Explanation
The quadratic formula is a powerful tool used to find the roots of any quadratic equation of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( x \) represents the variable.
The quadratic formula is expressed as:
\( a \) is the coefficient of the square term
\( b \) is the coefficient of the linear term
\( c \) is the constant term.
\( \sqrt{{b^2 - 4ac}} \): This is called the discriminant of the quadratic equation. It determines the number and nature of the roots of the equation. If the discriminant is positive, the equation has two distinct real roots. If the discriminant is zero, the equation has one real root (a repeated root). If the discriminant is negative, the equation has two complex roots (conjugate pairs).
\( -b \pm \sqrt{{b^2 - 4ac}} \): This expression represents the two possible solutions for \( x \). The \( \pm \) symbol indicates that there are two solutions, one obtained by adding the square root and the other by subtracting it. This is because a quadratic equation can have two real roots, one positive and one negative, or two complex roots.
\( 2a \): This is twice the coefficient of the square term ( \( x^2 \) ). It appears in the denominator to normalize the equation and ensure the solutions are correctly scaled.
Example: Solve the quadratic equation \( 3x^2 - 5x + 2 = 0 \) using the quadratic formula.
Solution:
Given the quadratic equation \( 3x^2 - 5x + 2 = 0 \), we can use the quadratic formula to find the roots \( x \).
First, let's identify the coefficients:
\( a = 3 \)
\( b = -5 \)
\( c = 2 \)
Now, plug these values into the quadratic formula:
\( x = \frac{{-(-5) \pm \sqrt{{(-5)^2 - 4 \cdot 3 \cdot 2}}}}{{2 \cdot 3}} \)
\( x = \frac{{5 \pm \sqrt{{25 - 24}}}}{{6}} \)
\( x = \frac{{5 \pm \sqrt{{1}}}}{{6}} \)
\( x = \frac{{5 \pm 1}}{{6}} \)
Now, we have two possible solutions:
- \( x = \frac{{5 + 1}}{{6}} = \frac{{6}}{{6}} = 1 \)
- \( x = \frac{{5 - 1}}{{6}} = \frac{{4}}{{6}} = \frac{{2}}{{3}} \)
Therefore, the roots of the quadratic equation \( 3x^2 - 5x + 2 = 0 \) are \( x = 1 \) and \( x = \frac{{2}}{{3}} \).
Deriving Quadratic Formula
Given a quadratic equation in the form \(ax^2 + bx + c = 0\), we want to express it in the form \((x - h)^2 = k\). Here's the step-by-step process:
1. Make the leading coefficient '1':
Divide both sides of the equation by 'a':
\(x^2 + \frac{b}{a}x + \frac{c}{a} = 0\)
2. Move the constant term to the right side:
Move the constant term \(\frac{c}{a}\) to the right side of the equation:
\(x^2 + \frac{b}{a}x = -\frac{c}{a}\)
3. Complete the square for the left side:
To complete the square, we need to add and subtract \(\left(\frac{b}{2a}\right)^2\) inside the bracket:
\(x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2 = -\frac{c}{a}\)
This can be rewritten as:
\(\left(x + \frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2 = -\frac{c}{a}\)
4. Simplify the equation:
\(\left(x + \frac{b}{2a}\right)^2 - \frac{b^2}{4a^2} = -\frac{c}{a}\)
5. Isolate the squared term:
Move the term \(-\frac{b^2}{4a^2}\) to the right side:
\(\left(x + \frac{b}{2a}\right)^2 = \frac{b^2}{4a^2} - \frac{c}{a}\)
6. Take the square root:
Take the square root of both sides:
\(x + \frac{b}{2a} = \pm \sqrt{\frac{b^2}{4a^2} - \frac{c}{a}}\)
7. Solve for `x`:
Subtract \(\frac{b}{2a}\) from both sides:
\(x = -\frac{b}{2a} \pm \sqrt{\frac{b^2}{4a^2} - \frac{c}{a}}\)
8. Simplify:
\(x = -\frac{b}{2a} \pm \frac{1}{2a} \sqrt{b^2 - 4ac}\)
This is the quadratic formula:
\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
The Discriminant
The nature of the roots of a quadratic equation depends on the discriminant, \(b^2 - 4ac\), where \(a\), \(b\), and \(c\) are the coefficients of the quadratic equation \(ax^2 + bx + c = 0\). The discriminant determines whether the equation has real roots, repeated roots, or complex roots. The discriminant, denoted by \(\Delta\), is the expression under the square root in the quadratic formula:
\( \Delta = b^2 - 4ac \)
`
1. Positive Discriminant (\(b^2 - 4ac > 0\)):
- If the discriminant is positive, the quadratic equation has two distinct real roots.
- Geometrically, this means the parabola representing the quadratic equation intersects the `x`-axis at two distinct points.
Example: Find the discriminant for quadratic equation \(2x^2 - 5x + 2 = 0\) and determine the nature of its roots.
Solution:
To find the discriminant for the quadratic equation \(2x^2 - 5x + 2 = 0\), we'll use the formula for the discriminant:
\( \Delta = b^2 - 4ac \)
Given the quadratic equation \(2x^2 - 5x + 2 = 0\), we can identify the coefficients \(a\), \(b\), and \(c\):
- \(a = 2\)
- \(b = -5\)
- \(c = 2\)
Now, substitute these values into the discriminant formula:
\( \Delta = (-5)^2 - 4 \cdot 2 \cdot 2 \)
\( \Delta = 25 - 16 \)
\( \Delta = 9 \)
The discriminant \(\Delta\) is `9`.
Now, let's determine the nature of the roots based on the value of the discriminant:
- Since \(\Delta > 0\), the quadratic equation \(2x^2 - 5x + 2 = 0\) has two distinct real roots.
- Geometrically, the graph of the quadratic equation will intersect the `x`-axis at two distinct points.
Therefore, the nature of the roots of the quadratic equation \(2x^2 - 5x + 2 = 0\) is that it has two distinct real roots.
2. Zero Discriminant (\(b^2 - 4ac = 0\)):
- If the discriminant is zero, the quadratic equation has one real root, which is a repeated root.
- Geometrically, this means the parabola representing the quadratic equation touches the `x`-axis at exactly one point.
Example: Find the discriminant for the quadratic equation \(x^2 - 6x + 9 = 0\) and determine the nature of its roots.
Solution:
To find the discriminant for the quadratic equation \(x^2 - 6x + 9 = 0\), we'll use the formula for the discriminant:
\( \Delta = b^2 - 4ac \)
Given the quadratic equation \(x^2 - 6x + 9 = 0\), we can identify the coefficients \(a\), \(b\), and \(c\):
- \(a = 1\)
- \(b = -6\)
- \(c = 9\)
Now, substitute these values into the discriminant formula:
\( \Delta = (-6)^2 - 4 \cdot 1 \cdot 9 \)
\( \Delta = 36 - 36 \)
\( \Delta = 0 \)
The discriminant \(\Delta\) is `0`.
Now, let's determine the nature of the roots based on the value of the discriminant:
- Since \(\Delta = 0\), the quadratic equation \(x^2 - 6x + 9 = 0\) has one real root (a repeated root).
- Geometrically, the graph of the quadratic equation touches the `x`-axis at one point (the vertex).
Therefore, the nature of the roots of the quadratic equation \(x^2 - 6x + 9 = 0\) is that it has one real root (a repeated root).
3. Negative Discriminant (\(b^2 - 4ac < 0\)):
- If the discriminant is negative, the quadratic equation has two complex roots, which are conjugate pairs.
- Geometrically, this means the parabola representing the quadratic equation does not intersect the `x`-axis; instead, it lies entirely above or below the `x`-axis.
Example: Determine the nature of the roots of the quadratic equation \(3x^2 + 4x + 7 = 0\).
Solution:
To determine the nature of the roots of the quadratic equation \(3x^2 + 4x + 7 = 0\), we need to analyze the discriminant. The discriminant (\(\Delta\)) is given by the formula:
\( \Delta = b^2 - 4ac \)
For the given quadratic equation \(3x^2 + 4x + 7 = 0\), we have:
- \(a = 3\)
- \(b = 4\)
- \(c = 7\)
Now, substitute these values into the discriminant formula:
\( \Delta = (4)^2 - 4 \cdot 3 \cdot 7 \)
\( \Delta = 16 - 84 \)
\( \Delta = -68 \)
The discriminant \(\Delta\) is negative (\(\Delta < 0\)).
Since the discriminant is negative, the quadratic equation \(3x^2 + 4x + 7 = 0\) has two complex roots. Geometrically, the graph of the quadratic equation does not intersect the `x`-axis.
Therefore, the nature of the roots of the quadratic equation \(3x^2 + 4x + 7 = 0\) is that it has two complex roots (conjugate pairs).
Sum and Product of the Roots
Let \(r_1\) and \(r_2\) be the roots of the quadratic equation \(ax^2 + bx + c = 0\).
Sum of Roots (\(r_1 + r_2\)):
The sum of the roots is given by:
\(r_1 + r_2 = -\frac{b}{a}\)
Example: Find the sum of the roots of the quadratic equation \(3x^2 + 4x - 5 = 0\).
Solution: Given the quadratic equation \(3x^2 + 4x - 5 = 0\), we have `a = 3`, `b = 4` and `c = -5`
\( r_1 + r_2 = -\frac{b}{a} = -\frac{4}{3} = -\frac{4}{3} \)
Therefore, the sum of the roots of the quadratic equation \(3x^2 + 4x - 5 = 0\) is \(-\frac{4}{3}\).
Product of Roots (\(r_1 \cdot r_2\)):
The product of the roots is given by:
\(r_1 \cdot r_2 = \frac{c}{a}\)
Example: Find the product of the roots of the quadratic equation \(2x^2 - 7x + 3 = 0\).
Solution: Given the quadratic equation \(2x^2 - 7x + 3 = 0\), we have `a = 2, b = -7` and `c = 3`
\( r_1 \cdot r_2 = \frac{c}{a} = \frac{3}{2} = \frac{3}{2} \)
Therefore, the product of the roots of the quadratic equation \(2x^2 - 7x + 3 = 0\) is \(\frac{3}{2}\).
Writing Quadratic Equations Using Roots
To write a quadratic equation using its roots, we can utilize the fact that if \(r_1\) and \(r_2\) are the roots of the quadratic equation \(ax^2 + bx + c = 0\), then the equation can be expressed as:
\( x^2 -(\text{sum of roots})x +\text{product of roots} = 0 \)
\( x^2 -(r_1+r_2)x +r_1\cdot r_2 = 0 \)
Given the roots \(r_1\) and \(r_2\), we can substitute them into the equation above to obtain the quadratic equation.
Example: Write a quadratic equation whose roots are `6` and `-3`.
Solution:
Roots: \( r_1 = 6 \) and \( r_2 = -3 \)
Substituting \(r_1\) and \(r_2\) values into \( x^2 -(r_1+r_2)x +r_1\cdot r_2 = 0 \), we get
\( x^2 - (6 + (-3))x + (6 \cdot (-3)) = 0 \)
\( x^2 - 3x - 18 = 0 \)
Hence the quadratic equation with roots `6` and `-3` is \( x^2 - 3x - 18 = 0 \).
Different Methods to Solve Quadratic Equations
There are several methods to solve quadratic equations. Here are the names of some commonly used methods:
- Factoring method
- Quadratic Formula
- Completing the Square method
- Graphical method
- Square Root method
Factoring Method
We employ the factoring method when the quadratic expression if factorable. We write the quadratic equation as a product of `2` linear factors. Next, we use the zero product property to set each factor equal to `0`. Finally, we solve each equation to find the roots.
Example: Solve \(x^2 - 5x + 6 = 0\) by factoring method.
Solution: To solve this quadratic equation by factoring, we'll look for two numbers that multiply to give the constant term `(6)` and add to give the coefficient of the linear term `(-5)`. In this case, those numbers are `-2 `and `-3` because `(-2) * (-3) = 6` and `(-2) + (-3) = -5`.
Now, we'll rewrite the quadratic equation by splitting the middle term `(-5x)` using these two numbers:
\(x^2 - 2x - 3x + 6 = 0\)
Next, we'll group the terms:
\((x^2 - 2x) + (-3x + 6) = 0\)
Now, we'll factor out the GCF (Greatest Common Factor) from each group:
\(x(x - 2) - 3(x - 2) = 0\)
Notice that we have a common factor of `(x - 2)` in both groups, so we can factor it out:
\((x - 2)(x - 3) = 0\)
Now, we have a product of two factors equal to zero. According to the zero-product property, if the product of two factors is zero, then at least one of the factors must be zero.
So, either \(x - 2 = 0\) or \(x - 3 = 0\).
Solving these linear equations:
- For \(x - 2 = 0\):
\(x = 2\) - For \(x - 3 = 0\):
\(x = 3\)
Therefore, the solutions to the quadratic equation \(x^2 - 5x + 6 = 0\) are \(x = 2\) and \(x = 3\).
Completing the Square Method
The method of completing the square is a technique used to rewrite a quadratic expression in the form of a perfect square trinomial. This method is particularly useful when solving quadratic equations or converting quadratic equations into vertex form.
Completing the square involves transforming a quadratic expression of the form \(ax^2 + bx + c\) into the form \((x + p)^2 + q\), where \(p\) and \(q\) are constants. This process effectively creates a perfect square trinomial, which can be easily factored and solved.
Steps to Complete the Square:
`1`. Ensure the Coefficient of \(x^2\) is 1: If the coefficient of \(x^2\) is not `1`, factor out the leading coefficient from the quadratic expression.
`2`. Move the Constant Term to the Right Side: Rewrite the quadratic expression so that the constant term (\(c\)) is isolated on one side of the equation, and the \(x\) terms are on the other side.
`3`. Create a Perfect Square Trinomial: Add and subtract \(\left(\frac{b}{2}\right)^2\) within the expression containing \(x\). This value is added to make a perfect square trinomial.
`4`. Factor the Perfect Square Trinomial: Write the perfect square trinomial as the square of a binomial.
`5`. Simplify and Solve: Rewrite the equation with the perfect square trinomial factored. Then, solve for \(x\) using techniques such as taking the square root or isolating \(x\).
Example: Solve the quadratic equation \(x^2 - 4x - 5 = 0\) using completing the square method.
- Ensure the coefficient of \(x^2\) is `1`: Already satisfied.
- Move the constant term to the right side: \(x^2 - 4x = 5\).
- Create a perfect square trinomial: Add and subtract \(\left(\frac{-4}{2}\right)^2 = 4\) within the expression containing \(x\): \(x^2 - 4x + 4 - 4 = 5\).
- Factor the perfect square trinomial: \((x - 2)^2 - 4 = 5\).
- Simplify and solve: \((x - 2)^2 = 9\). Taking the square root of both sides: \(x - 2 = \pm 3\). Therefore, \(x = 2 \pm 3\), yielding \(x = 5\) and \(x = -1\).
Solve Quadratic Equations by Graphing
The graphing method for solving quadratic equations involves graphing the quadratic function \(y = ax^2 + bx + c\) and finding the points where the graph intersects the `x`-axis, which represent the solutions to the equation \(ax^2 + bx + c = 0\). Here's an explanation of the graphical method:
`1`. Plot the Quadratic Function:
Begin by plotting the quadratic function \(y = ax^2 + bx + c\) on a cartesian coordinate system. The shape of the graph will be a parabola, which opens upwards if \(a > 0\) or downwards if \(a < 0\).
`2`. Locate the `x`-intercepts:
The `x`-intercepts of the graph are the points where the parabola intersects the `x`-axis. These points represent the solutions to the quadratic equation \(ax^2 + bx + c = 0\). To find these points, we look for the values of \(x\) where \(y = 0\).
`3`. Identify the Solutions:
The `x`-intercepts represent the solutions to the quadratic equation. If the graph intersects the `x`-axis at two distinct points, the equation has two real roots. If the graph touches the `x`-axis at a single point, the equation has one real root (a repeated root). If the graph does not intersect the `x`-axis at all, the equation has no real roots (the roots are complex).
`4`. Determine the Nature of the Roots:
Analyze the graph to determine the nature of the roots based on their behavior. If the graph is entirely above or below the `x`-axis, the equation has no real roots. If the graph intersects the `x`-axis, the equation has real roots, and the number of intercepts determines the number of real roots.
Example: Show the quadratic equation \(y = x^2 - 4x + 3\) by graphing method.
`1`. Plot the Quadratic Function:
Plot the function \(y = x^2 - 4x + 3\) on a graph. The graph will be a parabola that opens upwards.
`2`. Locate the `x`-intercepts:
Find the points where the graph intersects the `x`-axis.
`3`. Identify the Solutions:
The `x`-intercepts at \(x = 1\) and \(x = 3\) represent the solutions to the quadratic equation \(x^2 - 4x + 3 = 0\).
`4`. Determine the Nature of the Roots:
Since the graph intersects the `x`-axis at two distinct points, the equation has two real roots.
Square Root Method
The square root method is a technique used to solve quadratic equations by isolating the variable and then taking the square root of both sides of the equation. This method is particularly useful when the quadratic equation is in the form \(ax^2 = c\) or \((x-h)^2 = c\), where \(a\) and \(c\) are constants.
Below are the steps we follow when using the square root method:
`1`. Isolate the Variable: Move all terms except the variable term to the other side of the equation, leaving the variable term alone on one side.
`2`. Take the Square Root: Once the variable term is isolated, take the square root of both sides of the equation.
`3`. Consider Both Positive and Negative Roots: Remember that when taking the square root of both sides, you must consider both the positive and negative square roots on the side containing the constant term.
`4`. Solve for the Variable: After taking the square root, you'll have two equations, one with the positive square root and one with the negative square root. Solve both equations to find the possible values of the variable.
`5`. Check Solutions: Check the solutions by substituting them back into the original equation to ensure they satisfy the equation.
Example: Solve the quadratic equation \(x^2 = 25\) using the square root method.
Solution:
`1`. Isolate the Variable: The variable \(x^2\) is already isolated on one side of the equation.
`2`. Take the Square Root: Take the square root of both sides of the equation:
\( \sqrt{x^2} = \pm \sqrt{25} \)
`3`. Consider Both Positive and Negative Roots: Remember to include both the positive and negative square roots on the right side:
\( x = \pm 5 \)
`4`. Solve for the Variable: We have two solutions: \(x = 5\) and \(x = -5\).
`5`. Check Solutions: Substitute both \(x = 5\) and \(x = -5\) back into the original equation:
- For \(x = 5\): \(5^2 = 25\) (True)
- For \(x = -5\): \((-5)^2 = 25\) (True)
Practice Problems
Q`1`. What are the roots of the quadratic equation \(2x^2 - 5x + 3 = 0\)?
- \(x = 1, x = \frac{3}{2}\)
- \(x = 2, x = 4\)
- \(x = 3, x = 5\)
- \(x = -1, x = -3\)
Answer: a
Q`2`. Which is one of the values of \(x\) in the equation \(3x^2 + 6x - 9 = 0\)?
- \(x = -1\)
- \(x = 1\)
- \(x = 3\)
- \(x = -2\)
Answer: b
Q`3`. Which of the following quadratic equations has no real roots?
- \(x^2 + 4x + 4 = 0\)
- \(x^2 - 6x + 9 = 0\)
- \(x^2 + 2x + 5 = 0\)
- \(x^2 - 3x + 2 = 0\)
Answer: c
Q`4`. What are the solutions of \(x^2 - 9 = 0\)?
- \(x = 3\)
- \(x = -3\)
- \(x = \pm 3\)
- \(x = \pm 9\)
Answer: c
Q`5`. Which of the following equations represents a parabola that opens downwards?
- \(2x^2 + 4x + 1 = 0\)
- \(x^2 - 6x + 9 = 0\)
- \(3x^2 - 2x + 5 = 0\)
- \(-4x^2 + 2x + 6 = 0\)
Answer: d
Q`6`. The product of the roots of the quadratic equation \(4x^2 - 12x + 9 = 0\) is:
- \(-\frac{9}{4}\)
- \(-3\)
- \(3\)
- \(\frac{9}{4}\)
Answer: d
Q`7`. What is the discriminant of the quadratic equation \(x^2 + 5x + 4 = 0\)?
- \(\Delta = 5\)
- \(\Delta = 9\)
- \(\Delta = 1\)
- \(\Delta = -7\)
Answer: b
Q`8`. Which of the following is the correct quadratic formula?
- \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
- \(x = \frac{2a}{-b \pm \sqrt{b^2 - 4ac}}\)
- \(x = \frac{-b \pm \sqrt{b^2 + 4ac}}{2a}\)
- \(x = \frac{-b \pm \sqrt{b^2 - 2ac}}{a}\)
Answer: a
Q`9`. Which of the following quadratic equations has two distinct real roots?
- \(x^2 - 4x + 4 = 0\)
- \(x^2 + 2x + 1 = 0\)
- \(x^2 - 3x + 2 = 0\)
- \(x^2 + 4x + 4 = 0\)
Answer: c
Q`10`. What is the sum of the roots of the quadratic equation \(3x^2 - 4x - 5 = 0\)?
- `4`
- `-4`
- \(\frac{4}{3}\)
- \(\frac{-4}{3}\)
Answer: c
Frequently Asked Questions
Q`1`. What is a quadratic equation?
Answer: A quadratic equation is a second-degree polynomial equation in one variable, typically written in the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(a \neq 0\).
Q`2`. How do you solve a quadratic equation?
Answer: Quadratic equations can be solved using various methods such as factoring, the quadratic formula, completing the square, or graphical methods, depending on the form of the equation and the solver's preference.
Q`3`. What is the quadratic formula?
Answer: The quadratic formula provides the solutions (roots) to any quadratic equation \(ax^2 + bx + c = 0\). The formula is: \( x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} \)
Q`4`. What is the discriminant of a quadratic equation?
Answer: The discriminant (\(\Delta\)) of a quadratic equation \(ax^2 + bx + c = 0\) is the expression under the square root in the quadratic formula, given by \(\Delta = b^2 - 4ac\). It helps determine the nature of the roots of the quadratic equation.
Q`5`. What do you mean by the roots of a quadratic equation?
Answer: Roots are the values of \(x\) that satisfy the equation \(ax^2 + bx + c = 0\). The roots of a quadratic equation can be found using the quadratic formula, factoring, completing the square, or graphical method.
Q`6`. What are the properties of the roots of a quadratic equation?
Answer: The properties of the roots depend on the discriminant:
- If \(\Delta > 0\), the equation has two distinct real roots.
- If \(\Delta = 0\), the equation has one real root (a repeated root).
- If \(\Delta < 0\), the equation has two complex roots (conjugate pairs).
Q`7`. Can a quadratic equation have no real roots?
Answer: Yes, if the discriminant of a quadratic equation (\(b^2 - 4ac\)) is negative, the equation has no real roots. In this case, the roots are complex conjugates.
Q`8`. What is the relationship between the coefficients and roots of a quadratic equation?
Answer: Following is the relationship between the coefficients (\(a\), \(b\), and \(c\)) and the roots (\(r_1\) and \(r_2\)) of a quadratic equation:
- \(r_1 + r_2 = -\frac{b}{a}\)
- \(r_1 \cdot r_2 = \frac{c}{a}\)