To complete the square, make sure all the variable terms are on one side of the equation and all the constant terms are on the other side. Then, divide both sides of the equation by a so the coefficient of x^2 is 1.
See solution.
Practice makes perfect
We want use Quotient Property of Square Roots to derive the Quadratic Formula by solving the standard form of a quadratic equation. Let's start by rewriting the equation so that all of the terms with x are on one side of the equation and all of the constants are on the other side.
ax^2+bx+c=0 ⇔ ax^2+bx= - c
Following the given hint, we will begin by completing the square. Assuming that a ≠0, we will divide each side by a so the coefficient of x^2 will be 1.
ax^2+ bx= - c ⇔ 1x^2+ b/ax= - c/aTo complete the square, we will find the value of the linear coefficient divided by 2 and raised to the second power. Then we will add it to both sides of the equation. Usually, we describe this step as adding ( b2 )^2, but notice that this time our linear coefficient is not b but ba, so the expression that we add needs to be slightly different.
( b/a2 )^2=( b2 a )^2
Then, we will factor the trinomial on the left-hand side of the equation.
This expression is a perfectly valid formula for finding the solutions of a quadratic equation. However, since we want to derive the Quadratic Formula, we will simplify it further by using the Quotient Property of Square Roots and rewriting the right-hand side of the equation as one fraction.
