Now that you know how to factor quadratic expressions, it’s time to make that final jump to actually solving quadratic equations. When first discussing factoring, it was noted that when one side of the equation is equal to 0, you can make use of the rule that anything times 0 is 0. In the case of the equation (x − 5)(x + 10) = 0, you know that either (x − 5) = 0 or (x + 10) = 0, which means that x = 5 or x = −10.
The whole point of factoring quadratic equations is so that you can make use of this rule. Therefore, before you factor a quadratic expression, you must make sure that the other side of the equation equals 0.
Suppose you see an equation x2 + 10x = −21, and you need to solve for x. The x2 term in the equation should tell you that this is a quadratic equation, but it’s not yet ready to be factored. Before it can be factored, you have to move everything to one side of the equation. In this equation, the easiest way to do that is to add 21 to both sides, giving you x2 + 10x + 21 = 0. Now that one side equals 0, you are ready to factor.
The final term is positive, so you’re looking for two numbers to multiply to 21 and sum to 10. The numbers 3 and 7 fit the bill, so your factored form is (x + 3)(x + 7) = 0. That means that x = −3 or x = −7.
Now you know all the steps to successfully factor and solve quadratic equations.
Solve the following quadratic equations.
x2 − 3x + 2 = 0
x2 + 2x − 35 = 0
x2 − 15x = −26