Problem 95 Solve the equation. $$ (x-7)... [FREE SOLUTION]

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Chapter 1: Problem 95

Solve the equation. $$ (x-7)(x+2)=x^{2}+4 x+13 $$

Short Answer

Expert verified

x = -3

Step by step solution

Expand the left side

Start by expanding the left side of the equation (x-7)(x+2) = x^2 + 2x - 7x - 14 = x^2 - 5x - 14

Set up the equation

After expanding, set the equation equal to the right side: x^2 - 5x - 14 = x^2 + 4x + 13

Subtract x^2 from both sides

Since both sides contain x^2, subtract x^2 from both sides to simplify: -5x - 14 = 4x + 13

Isolate the variable x

Move all terms involving x to one side of the equation. Add 5x to both sides: -14 = 9x + 13

Solve for x

Isolate x by subtracting 13 from both sides -14 - 13 = 9x -27 = 9x Finally, divide both sides by 9 to find x: x = -3

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Key Concepts

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

Expanding Expressions

To solve quadratic equations, a crucial first step is expanding expressions. This mainly involves multiplying terms to simplify the equation. For the equation $(x - 7)(x + 2) = x^2 + 4x + 13$, we start by expanding the left side.
When expanding expressions, each term in the first parenthesis multiplies with each term in the second.
Specifically:
* Multiply $(x * x) = x^2$
* Multiply $(x * 2) = 2x$
* Multiply $(-7 * x) = -7x$
* Multiply $(-7 * 2) = -14$
Combining these, the left side becomes $(x^2 + 2x - 7x - 14 = x^2 - 5x - 14)$. This step simplifies the initial equation, laying the groundwork for isolating variables later.

Isolating Variables

After expanding expressions, we need to isolate the variable. In the equation $(x^2 - 5x - 14 = x^2 + 4x + 13)$, both sides have the term $x^2$.
Subtract $x^2$ from both sides to eliminate it, simplifying to $(-5x - 14 = 4x + 13)$. This step removes the squared terms.
The next focus is on isolating $(x)$. Move all x terms to one side by adding $(5x)$ to both sides, which results in $(-14 = 9x + 13)$.
This step consolidates x terms, making it easier to solve for the variable.

Basic Algebra Steps

The final phase involves basic algebra to solve for x. With $( -14 = 9x + 13)$, the goal is to isolate $(x)$.
Subtract $(13)$ from both sides to get $-27 = 9x$.
This aligns one side with just the $x$ term multiplied by a coefficient.
Finally, divide both sides by $(9)$ to isolate the variable: $ x = -3 $.
These algebraic steps ensure that x is isolated and the equation is solved thoroughly and correctly.

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91影视

Solve the equation. $$ (x-7)(x+2)=x^{2}+4 x+13 $$

Short Answer

Step by step solution

Expand the left side

Set up the equation

Subtract x^2 from both sides

Isolate the variable x

Solve for x

Key Concepts

Expanding Expressions

Isolating Variables

Basic Algebra Steps

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