How to Solve Quadratic Equations Using Four Different Methods

A quadratic equation is an equation that can be written in the form:ax^2 + bx + c = 0, where a, b and c are real numbers and a is not 0.Quadratic equations have two solutions, which are not necessarily unique.Algebra introduces quadratic equations and possible ways to solve them. This article will address four different methods for solving quadratic equations: factoring, completing the square, using the quadratic formula and using Microsoft Excel.The first step in each method is to write the equation in standard quadratic equation form, ax^2 + bx + c = 0.

Difficulty: Moderately Challenging

Instructions

Things You'll Need:

• Calculator
• Microsoft Excel

1. Solve by factoring:Example:x^2 = 9Write the equation in standard quadratic form by subtracting 9 from both sides: x^2 - 9 = 0Factor to write the polynomial as a product: (x + 3)(x - 3) = 0Set each factor equal to 0: (x + 3) = 0 or (x - 3) = 0Solve each factor: x = -3 or x = 3

2. Solve by completing the square:Example:x^2 = 9Write in standard quadratic equation form by subtracting 9 from both sides: x^2 - 9 = 0Apply the square root property: x = +/- square root of 9Solve the square root: x = +/- 3

3. Solve by using the quadratic formula:Example:3x^2 + 16x + 5 = 0This example is already written in the standard quadratic equation form; therefore, we know that a = 3, b = 16 and c = 5.Substitute the values for a, b and c into the quadratic formula:x = (-b +/- square root(b^2 - 4ac)) / (2a)x = (-16 +/- square root(16^2 - 4(3)(5))) / (2(3))x = (-16 +/- square root(256 - 60)) / 6x = (-16 +/- square root(196)) / 6x = (-16 +/- 14) / 6x = (16 - 14) / 6 or x = (16 + 14) / 6x = -1/3 or x = -5Apply the square root property: x = +/- square root of 9Solve the square root: x = +/- 3

4. Solve by using Microsoft Excel:Example:3x^2 + 16x + 5 = 0This example is already written in the standard quadratic equation form; therefore, we know that a=3, b=16 and c=5.In ExcelColumn A = aColumn B = bColumn C = cColumn D = the first solution for x=((-B2)+SQRT((B2*B2)-4*A2*C2))/(2*A2)Column E = the second solution for x=((-B2)-SQRT((B2*B2)-4*A2*C2))/(2*A2)Substitute the values for a, b and c into the quadratic formula:x = (-b +/- square root(b^2 - 4ac)) / (2a)x = (-16 +/- square root(16^2 - 4(3)(5))) / (2(3))x = (-16 +/- square root(256-60)) / 6x = (-16 +/- square root(196)) / 6x = (-16 +/- 14) / 6x = (16 - 14) / 6 or x = (16 + 14) / 6x = -1/3 or x = -5

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