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# Solving Quadratic Equations by Factoring

## Solving an Equation of the Form ax2 + bx + c = 0 by Factoring

The number 0 has several special properties, including this one:

â€¢ The product of 0 and any number is 0. For example, 2 Â· 0 = 0.

Conversely, if the product of two real numbers is 0, then one or both must be 0. This statement is known as the Zero Product Property.

Property â€” Zero Product Property

English If the product of two numbers is 0, then one or both of the numbers is 0.

Algebra If a and b are real numbers, and if a Â· b = 0, then a = 0 or b = 0 or both a and b are equal to 0.

Example If (x + 2)(x + 3) = 0, then x + 2 = 0 or x + 3 = 0.

The Zero Product Property is useful for solving certain types of equations.

Example

Solve: (x - 3)(x + 5) = 0

Solution

 The product of (x - 3) and (x + 5) is zero. (x - 3)(x + 5) = 0 Use the Zero Product Property to write two separate equations. Solve each equation. x - 3 = 0 x = 3 or or x + 5 = 0x = -5
Thus, the original equation has two solutions: 3 and -5.

The solutions may be checked by substituting each value of x into the original equation and simplifying.

 Check x = 3. Check x = -5. (x - 3)(x + 5) = 0 (x - 3)(x + 5) = 0 IsIs Is (3 - 3)(3 +5) (0)(8) 0 = 0 ?= 0 ? = 0 ? Yes IsIs Is (-5 - 3)(-5 + 5) (-8)(0) 0 = 0 ?= 0 ? = 0 ? Yes

A quadratic equation is an equation that can be written in the form ax2 + bx + c = 0, where a, b, and c are real numbers and a 0.

A quadratic equation written in this form is said to be in standard form.

 Now we will rewrite this equation. We multiply the binomials. We combine like terms. (x - 3)(x + 5) = 0x2 + 5x - 3x - 15 = 0x2 + 2x - 15 = 0
This quadratic equation is written in standard form.

Notice that the terms on the left side of ax2 + bx + c = 0 are arranged in descending order by degree. The right side of the equation is zero.

Quadratic equations are also called second-degree equations because the degree of the polynomial, ax2 + bx + c, is 2.

We will use the Zero Product Property to solve some quadratic equations by factoring.

Procedure â€” To Solve a Quadratic Equation By Factoring

Step 1 Write the quadratic equation in the form ax2 + bx + c = 0.

Step 2 Factor the polynomial.

Step 3 Use the Zero Product Property.

Step 4 Solve each equation.