# Solving Quadratic Equations by Completing the Square

February 22, 2023

Completing the square is a technique that can be used to rewrite certain quadratic equations as perfect square trinomials, which can then be factored and solved using the square root property. Solving quadratic equations by completing the square is useful for equations that cannot be easily factored or solved using other methods.

## Steps for Solving Quadratic Equations by Completing the Square Method

Completing the square is a technique that can be used to solve certain types of equations. Here are the steps on how to solve equations by completing the square:

- Rewrite the equation in the form \( x^2 + bx = c \).
- Add the term \( (\frac{b}{2})^2 \) to both sides of the equation. This term is needed to complete the square.
- Factor the resulting expression as a perfect square trinomial: \( (x + \dfrac{b}{2})^2 = d \).
- Take the square root of both sides and solve for x. This step requires using the square root property: \( x + \dfrac{b}{2} = \pm \sqrt{d} \), so \( x = – \dfrac{b}{2} \pm \sqrt{d} \).

Visually, completing the square for \(x^2 + bx \) involves adding \( (\frac{b}{2})^2 \) to both sides. The general process involves setting one side of the equation to zero, making the leading coefficient one, finding \( (\frac{b}{2})^2 \), adjusting the equation accordingly, factoring the perfect square trinomial, taking the square root of both sides and solving for x.

## How to complete the square using an example

- To determine the constant term of a perfect square expression beginning with \(x^2 + 6x \), we can use the technique of completing the square. Assuming that the expression can be factored as \((x + a)^2 \), where a is a constant, we can expand this expression to obtain \( x^2 + 2ax + a^2 \).
- From this expanded form, we know that the coefficient of x in the original expression, which is 6, should equal 2a. Therefore, we can solve for a by dividing 6 by 2, giving a value of 3.
- To find the constant term that needs to be added to form a perfect square, we can take \( a^2 \), equal to \( 3^2 \) or 9. So the constant term that needs to be added is 9. Therefore, the perfect square expression beginning with \( x^2 + 6x \) is \( (x + 3)^2 \).

## Quadratic Equations: Solve by Completing the Square

**Example: **\( x^2 − 8x + 13 = 0 \)

To solve the quadratic equation \( x^2 − 8x + 13 = 0 \) using the complete square method, we can follow the following steps:

- Move the constant term (in this case, 13) to the right-hand side of the equation: \( x^2 − 8x = -13 \)
- Add and subtract the square of half the coefficient of x (in this case, \( – \dfrac{8}{2} = -4 \) on the left-hand side of the equation: \( x^2 − 8x + (-4)^2 – (-4)^2 = -13 \)
- Simplify the left-hand side by combining like terms: \( (x – 4)^2 – 3 = 0 \)
- Add 3 to both sides of the equation: \( (x – 4)^2 = 3 \)
- Take the square root of both sides: \( x – 4 = \pm \sqrt{3} \)
- Add 4 to both sides: \( x = 4 \pm \sqrt{3} \)
- Therefore, the solutions to the quadratic equation \( x^2 − 8x = -13 \) using the complete square method are \( x = 4 + \sqrt{3} \) and \( x = 4 – \sqrt{3} \).

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