Problem 32 Solve each equation. $$ (x+8... [FREE SOLUTION]

Chapter 5: Problem 32

Solve each equation. $$ (x+8)(x-2)=-21 $$

Short Answer

Expert verified

The solution is x = -1 and x = -5.

Step by step solution

- Expand the Equation

First, expand the left side of the equation by using the distributive property: (x+8)(x-2) = x^2 - 2x + 8x - 16 = x^2 + 6x - 16.

- Set the Equation to Zero

Rewrite the equation with the right side set to zero: x^2 + 6x - 16 = -21 Add 21 to both sides to move all terms to one side: x^2 + 6x - 16 + 21 = 0 This simplifies to: x^2 + 6x + 5 = 0.

- Factor the Quadratic Equation

Factor the quadratic equation: (x + 1)(x + 5) = 0.

- Solve for x

Set each factor equal to zero and solve for x: x + 1 = 0 x = -1 x + 5 = 0 x = -5.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

factoring quadratics

When solving quadratic equations, factoring is a powerful tool. Quadratic equations take the form $ax^2 + bx + c = 0$. Factoring involves expressing the quadratic as a product of two binomials. For example, $x^2 + 6x + 5$ can be written as $(x + 1)(x + 5)$.
Factoring is often easier if you start by identifying pair of factors of the constant term that add up to the coefficient of the linear term.
Practicing factoring can make it much easier to handle quadratic equations efficiently.

distributive property

The distributive property is used to expand expressions. This means multiplying each term inside a parenthesis by the term outside it: $a(b + c) = ab + ac$.
In our example, $ (x+8)(x-2)$, we apply the distributive property as follows: multiply x by x and -2, and then 8 by x and -2.
Expanding $ (x+8)(x-2) = x^2 - 2x + 8x - 16 = x^2 + 6x - 16$.
The result is a standard quadratic equation, which can then be further solved.

setting equations to zero

To solve quadratic equations, setting them to zero simplifies the process. Rewrite the given equation to have one side equal to zero.
In our example, transform $ x^2 + 6x - 16 = -21$ by adding 21 to both sides: $ x^2 + 6x - 16 + 21 = 0 $, simplifying to $ x^2 + 6x + 5 = 0 $.
This form, where one side is zero, allows you to use various methods such as factoring or the quadratic formula to find solutions.

solving for x

After factoring the quadratic equation, set each factor equal to zero to solve for the variable.
For instance, if you have $ (x + 1)(x + 5) = 0 $, set each factor equal to zero: $ x + 1 = 0 $ and $ x + 5 = 0 $.
Solving these gives \ x = -1 \ and \ x = -5 \ respectively.
Thus, the solutions to the equation are \ x = -1 \ and \ x = -5 \. This step-by-step process ensures all possible solutions are found.

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91影视

Solve each equation. $$ (x+8)(x-2)=-21 $$

Short Answer

Step by step solution

- Expand the Equation

- Set the Equation to Zero

- Factor the Quadratic Equation

- Solve for x

Key Concepts

factoring quadratics

distributive property

setting equations to zero

solving for x

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