The square root method can be used for solving quadratic equations in the form "x² = b." This method can yield two answers, as the square root of a number can be a negative or a positive number. If an equation can be expressed in this form, it can be solved by finding the square roots of x.
Put the Equation Into the Proper Form
In the equation x² - 49 = 0, the second element on the left side (-49) must be removed to isolate x². This is easily accomplished by adding 49 to both sides of the equation. It is important to remember to always apply changes like this to both sides of the equal sign or you will get an incorrect answer. x² - 49 (+ 49) = 0 (+ 49) yields an equation in the proper form for the square root method: x² = 49.
Find the Roots
x² is made up of an element (x) which has been squared, or multiplied by itself (x · x). In other words, finding the square root is finding the number (x or -x) that is the root of the squared number. In the equation x² = 49, √49 = +/- 7, yielding the final answer x = +/- 7.
Isolate the Square
Sometimes you may be given an equation to solve by this method that is in the form ax² = b. In this case, you can isolate x² by multiplying both sides of the equation by the reciprocal of "a." The reciprocal of "a" is 1/a, and the product of these terms equals 1. If you have a fraction, such as 3/4, simply turn the fraction upside down to get its reciprocal: 4/3.
Example With Reciprocal
In the equation 6x² = 72, multiplying both sides of the equation by the reciprocal of 6, or 1/6, will convert it to the proper form for solving by this method. The equation (1/6)6x² = 72(1/6) works out to x² = 12. X then is equal to √12. You can then factor 12: 12 = 2 · 2 · 3, or 2² · 3. Remembering that either the positive or negative square root could be the answer yields the final answer: x = +/- 2√3.