How to Solve Quadratics by Square Roots
Quadratics -- equations with binomials -- arise in many practical applications. There are three standard ways to solve quadratics: by factoring, by using the quadratic formulas and by completing the square. The problem with factoring is that not all binomials can be factored. The quadratic formula will always work, but it is not always the simplest way to find a solution. Completing the square is often the simplest way to find the solution.
Other People Are Reading
Instructions
-
-
1
Set up the quadratic so you can write one side of the equation as a perfect square. In this form, you can take the square root of both sides of the equation and have a monomial on one side of the equation and a number on the other side of the equation. These equations are easy to solve. The first step in this process is to put all the terms with variables on one side of the equation and the constant term on the other side.
-
2
Add the same number to both sides of the equation so the binomial is a perfect square. For example, if you start with the quadratic X^2 - 2X - 63 = 0, start by putting all the variables on one side and the constant term on the other side to get X^2 - 2X = 63. Add the right amount to both sides so the binomial side is a perfect square. There is a formula to compute this amount: half the coefficient of the second term squared. The coefficient is 2 so (1/2(2))^2 = (1)^2 = 1 is added to both sides to get X^2 - 2X + 1 = 64, which is the same as (X - 1)^2 = 64.
-
-
3
Take the square root of both sides of the equations once the binomial is a perfect square. The constant term on the other side of the equation will have two square roots -- one positive, the other negative -- so there will be two equations. For example, if (X - 1)^2 = 64, the two equations will be X - 1 = 8 and X - 1 = -8, so X = 9 and X = -7. Check the solutions in the original quadratic: (9)^2 - 2(9) - 63 = 81 - 18 -63 = 0 and (-7)^2 - 2(-7) - 63 = 49 + 14 - 63 = 0.
-
1
Tips & Warnings
If the leading coefficient is not one, you should divide through the equation by this coefficient. Completing the square works only if the leading coefficient is one.
When you are ready to take the square root of both sides, it is entirely possible that the constant on the right hand side of the equation will be negative. If you are not allowed to use complex numbers -- if you have not studied complex numbers yet -- you can conclude that this polynomial cannot be solved.
References
Resources
- Photo Credit Jupiterimages/Photos.com/Getty Images
