# Solve The Quadratic Equation By Completing The Square Worksheet

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Solve the quadratic equation by completing the square transforms $$ax^2 + bx + c = 0$$ into $$a(x - h)^2 + k = 0$$ by adjusting the equation to isolate $$x$$. This method involves adding and subtracting a constant to complete the square on $$x$$, simplifying the equation for solving by taking the square root.

Algebra 2

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

• Clarifies completing the square method for quadratic equations.
• Develops critical thinking and algebraic skills.
• Enhances understanding of quadratic graphs.
• Strengthens equation manipulation abilities.
• Applies math concepts to real-world scenarios.
• Prepares for advanced math topics.
• Encourages self-paced learning.

## How to Solve the Quadratic Equation by Completing the Square?

• Ensure the equation is in the form  $$ax^2 + bx + c = 0$$.
• If the coefficient $$a$$ of $$x^2$$ is not 1, divide the entire equation by $$a$$.
• Move the constant term $$c$$ to the other side of the equation.
• Take half of the coefficient of$$x$$ (which is \frac{b}{2}), square it \left(\frac{b}{2}\right)^2,  and add and subtract this square inside the equation.
• Rewrite the left side of the equation as a perfect square trinomial, and simplify the equation.
• Take the square root of both sides of the equation, and solve for $$x$$.

## Solved Example

Q. Solve by completing the square.$\newline$$x^2 + 8x = 29$$\newline$Write your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.$\newline$$x =$_____ or $x =$_____
Solution:
1. Write Equation Form: Write the equation in the form of $x^2 + bx = c$. We have the equation $x^2 + 8x = 29$.
2. Complete Square: Complete the square by adding $\left(\frac{b}{2}\right)^2$ to both sides of the equation.$\newline$Since $\left(\frac{8}{2}\right)^2=16$, we add $16$ to both sides to complete the square.$\newline$$x^2 + 8x + 16 = 29 + 16$$\newline$$x^2 + 8x + 16 = 45$
3. Factor Left Side: Factor the left side of the equation.$\newline$The left side is a perfect square trinomial.$\newline$$(x + 4)^2 = 45$
4. Take Square Root: Take the square root of both sides of the equation.$\newline$$\sqrt{(x + 4)^2} = \pm\sqrt{45}$$\newline$$x + 4 = \pm\sqrt{45}$
5. Solve for x: Solve for x by isolating the variable.$\newline$Subtract $4$ from both sides of the equation.$\newline$$x = -4 \pm \sqrt{45}$
6. Simplify Square Root: Simplify the square root and round to the nearest hundredth if necessary.$\newline$$\sqrt{45}$ is approximately $6.71$.$\newline$$x = -4 \pm 6.71$
7. Find Values of x: Find the two values of x.$\newline$$x \approx -4 + 6.71$ implies $x \approx 2.71$.$\newline$$x \approx -4 - 6.71$ implies $x \approx -10.71$.$\newline$Values of x: $2.71$, $-10.71$

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