Ch. 1 - Equations and Inequalities

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Blitzer 8th Edition

Ch. 1 - Equations and Inequalities

Problema 58

Capitolo 2, Problema 58

Solve each equation in Exercises 58–59 by factoring. 2x^2 +15x = 8

Guida verificata passo dopo passo

Rewrite the equation in standard form by moving all terms to one side of the equation, so it becomes:

2x^2 + 15x - 8 = 0

Factor the quadratic equation. Look for two numbers that multiply to

2 	imes -8 = -16

and add to

15

. These numbers are

16

and

-1

Rewrite the middle term

15x

using the two numbers found:

2x^2 + 16x - x - 8 = 0

Group the terms into two pairs and factor each group:

(2x^2 + 16x) - (x + 8) = 0

, which simplifies to

2x(x + 8) - 1(x + 8) = 0

Factor out the common binomial factor

(x + 8)

(2x - 1)(x + 8) = 0

. Set each factor equal to zero to solve for

x

2x - 1 = 0

and

x + 8 = 0

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Factoring Quadratic Equations

Factoring is the process of breaking down a quadratic equation into simpler expressions that can be multiplied to yield the original equation. In the case of a quadratic in the form ax^2 + bx + c = 0, we look for two numbers that multiply to ac and add to b. This technique simplifies solving the equation by setting each factor to zero.

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Solving Quadratic Equations by Factoring

Setting the Equation to Zero

To solve a quadratic equation by factoring, it is essential to first rearrange the equation so that one side equals zero. This is done by moving all terms to one side, resulting in a standard form of ax^2 + bx + c = 0. This step is crucial because it allows us to apply the Zero Product Property, which states that if the product of two factors equals zero, at least one of the factors must be zero.

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Finding Zeros & Their Multiplicity

Zero Product Property

The Zero Product Property is a fundamental principle in algebra that states if the product of two or more factors equals zero, then at least one of the factors must be zero. This property is used after factoring a quadratic equation, allowing us to set each factor equal to zero and solve for the variable. It is a key step in finding the solutions to the original equation.

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