9-4 Skills Practice Solving Quadratic Equations by Completing the Square

What is the method for solving quadratic equations by completing the square?

The method for solving quadratic equations by completing the square involves converting the equation into a perfect square trinomial. By doing this, we can find the roots of the equation. Completing the square simplifies the process of finding the roots compared to using the quadratic formula.

Completing the Square Method

Completing the square method is a technique used to solve quadratic equations by converting them into a perfect square trinomial. This method involves adding and subtracting a constant to form a square of a binomial. Step 1: Start with the quadratic equation in the form \( ax^2 + bx + c = 0 \). Step 2: Move the constant term to the other side of the equation: \( ax^2 + bx = -c \). Step 3: Divide the coefficient of the \( x^2 \) term by 2 and square it: \( \left(\frac{b}{2}\right)^2 \). Step 4: Add and subtract the square obtained in Step 3 to the equation: \( ax^2 + bx + \left(\frac{b}{2}\right)^2 = -c + \left(\frac{b}{2}\right)^2 \). Step 5: Factor the trinomial on the left side of the equation: \( a\left(x+\frac{b}{2a}\right)^2 = -c + \left(\frac{b}{2}\right)^2 \). Step 6: Solve for \( x \) by taking the square root of both sides and isolating \( x \). By completing the square, we simplify the process of finding the roots of a quadratic equation. This method can be especially helpful when dealing with equations that are not in standard form or when finding the domain and range of a function in vertex form. Converting the equation into a perfect square trinomial allows us to solve it efficiently as \( x \) appears only once in the equation.
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