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Solving a quadratic equation using the quadratic formula means finding the values of $$x$$ that satisfy $$ax^2 + bx + c = 0$$ by using the formula x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. This formula helps pinpoint where the equation equals zero on a graph, giving precise solutions based on the equation's coefficients $$a$$, $$b$$, and $$c$$. Use these worksheets to enhance your problem solving skills.

Algebra 2

## How Will This Worksheet on “Solve the Quadratic Equation by Quadratic Formula” Benefit Your Student's Learning?

• Reinforces understanding of the quadratic formula.
• Provides extensive practice in solving quadratic equations.
• Bridges theoretical math concepts with real-world applications.
• Improves algebraic manipulation skills.
• Enhances graphing and visual interpretation abilities.
• Prepares students effectively for exams and assessments.
• Encourages independent problem-solving.
• Develops critical thinking through logical problem-solving steps.

1. Ensure the equation is in the form $$ax^2 + bx + c = 0$$ and identify the values of $$a$$, $$b$$, and $$c$$.

2. Substitute the values of $$a$$, $$b$$, and $$c$$ into the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

3. Compute $$b^2 - 4ac$$, known as the discriminant.

4. Depending on the value of the discriminant:

• If $$b^2 - 4ac > 0$$, there are two real solutions.
• If $$b^2 - 4ac = 0$$, there is one real solution (a repeated root).
• If $$b^2 - 4ac < 0$$, there are two complex solutions.

5. Substitute the discriminant value into the formula and calculate $$x$$ using both the plus and minus signs in the formula to find the solutions.

## Solved Example

Q. Solve using the quadratic formula.$\newline$$2d^2 - 8d + 6 = 0$$\newline$Write your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.$\newline$d = _____ or d = _____
Solution:
1. Identify values: Identify the values of $a$, $b$, and $c$ in the quadratic equation $2d^2 - 8d + 6 = 0$ by comparing it to the standard form $ax^2 + bx + c = 0$.$\newline$$a = 2$$\newline$$b = -8$$\newline$$c = 6$
2. Substitute in formula: Substitute the values of $a$, $b$, and $c$ into the quadratic formula to find $d$.$d = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$d = \frac{-(-8) \pm \sqrt{(-8)^2 - 4\cdot2\cdot6}}{2\cdot2}$
3. Simplify terms: Simplify the terms under the square root and the constants outside the square root. $\newline$$d = \frac{8 \pm \sqrt{64 - 48}}{4}$$\newline$$d = \frac{8 \pm \sqrt{16}}{4}$
4. Calculate square root: Calculate the square root of $16$ and simplify the expression further.$\newline$$d = \frac{(8 \pm 4)} {4}$$\newline$Identify the two possible values for $d$.$\newline$$d = \frac{(8 + 4)} {4}$or $d = \frac{(8 - 4)} {4}$
5. Identify possible values: Solve for the two values of $d$.$\newline$$d = \frac{12}{4}$ or $\newline$$d = \frac{4}{4}$$\newline$$d = 3$, or $d = 1$

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