# Solve Quadratic Equations by Factoring

This is a tutorial on how to solve quadratic equations by factoring. There is also Factor Quadratic Expressions - Step by Step Calculator

 Example 1 : Solve the following quadratic equation. x 2 - 3x = 0 Solution to Example 1: Given x 2 - 3x = 0 Factor x out in the expression on the left. x (x - 3) = 0 For the product x (x - 3) to be equal to zero we nedd to have x = 0 or x - 3 = 0 Solve the above simple equations to obtain the solutions. x = 0 or x = 3 As an exercise, check that x = 0 and x = 3 are solutions to the given equation. Example 2 : Solve the quadratic equation given below x 2 - 5 x + 6 = 0 Solution to Example 2: To factor the expression on the left, we need to write x 2 - 5 x + 6 in the form factored: x 2 - 5 x + 6 = (x + a)(x + b) so that the sum of a and b is -5 and their product is 6. The numbers that satisfy these conditions are - 2 and - 3. Hence x 2 - 5 x + 6 = (x - 2)(x - 3) Substitute into the original equation and solve. (x - 2)(x - 3) = 0 (x - 2)(x - 3) is equal to zero if x - 2 = 0 or x - 3 = 0 Solve the above equations to obtain two solutions to the given equation. x = 2 or x = 3 As an exercise, check that x = 0 and x = 3 are solutions to the given equation. Example 3: Solve the following equation 2 x 2 + x - 21 = 0 Solution to Example 3: We first try to write 2 x 2 + x - 21 in the factored form 2 x 2 + x - 21 = (2x + a)(x + b) Such that the product a b is equat to - 21 and a + 2 b = 1 two pairs of numbers gives a product of - 21: either -3 and 7 or 3 and -7. After some trial exercises it found that 2 x 2 + x - 21 may be factored as follows: 2 x 2 + x - 21 = (2x + 7)(x - 3) We now substitute into the original equation (2x + 7)(x - 3) = 0 and solve the following simpler equations 2x + 7 = 0 x - 3 = 0 to obtain x = - 7 / 2 or x = 3 As an exercise, check that x = 0 and x = 3 are solutions to the given equation. Example 4: Solve the following equation (x - 1)(x + 1 / 2) = - x + 1 Solution to Example 4: At first we might be tempted into expanding the left side of the equation. However after examination of the right side, the above equation may be written as: (x - 1)(x + 1 / 2) = - (x - 1) Write the equation with the right side equal to zero. (x - 1)(x + 1 / 2) + (x - 1) = 0 We now factor (x - 1) out. (x - 1)(x + 1 / 2 + 1) = 0 and solve the following simpler equations x - 1 = 0 x + 3 / 2 = 0 to obtain x = 1 or x = - 3 / 2 More references and links to quadratic equations. Quadratic Equations Calculator and Solver. More references and links on how to Solve Equations, Systems of Equations and Inequalities.