Problem 40 For the following exercises, sol... [FREE SOLUTION]

91影视

College Algebra with Corequisite Support

Jay Abramson

$Math Studyset 91影视 Explanations$ Math

2020 Edition

Chapter 2: Problem 40

For the following exercises, solve the equation by identifying the quadratic form. Use a substitute variable and find all real solutions by factoring. $$ (x+1)^{2}-8(x+1)-9=0 $$

Short Answer

Expert verified

The real solutions are $x = 8$ and $x = -2$.

Step by step solution

Identify the Substitution Variable

Recognize the expression $(x+1)$ appearing repeatedly in the equation. Let $u = x+1$. Substitute $u$ into the equation to rewrite $(x+1)^2 - 8(x+1) - 9 = 0$ as $u^2 - 8u - 9 = 0$.

Rewrite the Equation in Terms of the New Variable

The equation $(x+1)^2 - 8(x+1) - 9 = 0$ now simplifies to the quadratic equation $u^2 - 8u - 9 = 0$. This equation is in quadratic form and ready for factoring.

Factor the Quadratic Equation

Factor the quadratic equation $u^2 - 8u - 9$. Find two numbers that multiply to $-9$ and add up to $-8$. These numbers are $-9$ and $+1$. Thus, the equation factors as $(u - 9)(u + 1) = 0$.

Solve for the Substitute Variable

Set each factor from $(u - 9)(u + 1) = 0$ equal to zero and solve for $u$.- For $u - 9 = 0$, we get $u = 9$.- For $u + 1 = 0$, we get $u = -1$.

Substitute Back to Original Variable

Recall that $u = x + 1$. Substitute back to find the values of $x$.- If $u = 9$, then $x + 1 = 9$, giving $x = 8$.- If $u = -1$, then $x + 1 = -1$, giving $x = -2$.

Verify the Real Solutions

Check each solution in the original equation to ensure they satisfy it.- For $x = 8$, substitute into the original equation: $(8 + 1)^2 - 8(8 + 1) - 9 = 0$, which simplifies to $81 - 72 - 9 = 0$.- For $x = -2$, substitute into the original equation: $(-2 + 1)^2 - 8(-2 + 1) - 9 = 0$, which simplifies to $1 + 16 - 9 = 0$. Both calculations confirm the solutions are correct.

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Key Concepts

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

Factoring

Factoring is a fundamental technique used to solve quadratic equations. Essentially, it involves rewriting the equation as a product of two smaller expressions set to zero. Here are the basics of how it works:

Identify a quadratic equation of the form $ ax^2 + bx + c = 0 $.
Find two numbers that multiply to the constant term $ c $ and add up to the coefficient $ b $.
Use these numbers to break down the middle term (the one with $ x $) and rewrite the equation.
Factor by grouping and simplify the expression to its factored form.

This approach breaks down complex expressions into simpler, solvable components. In our exercise, the equation $ u^2 - 8u - 9 = 0 $ was factored as $(u - 9)(u + 1) = 0$, which simplifies finding solutions.

Substitution Method

The substitution method simplifies solving complex equations by introducing a new variable. This method is especially useful when the equation contains a recurring expression. Here鈥檚 how it works:

Identify the part of the equation that repeats. Assign it a substitute variable.
Rewrite the equation using this new variable and solve it as a standard quadratic equation.
Once solved, substitute back the original terms for your new variable to find the original variable's solutions.

In the given problem, the repeated expression $ (x+1) $ was replaced with $ u $, turning the equation into $ u^2 - 8u - 9 = 0 $. This transformation allowed us to factor and solve the equation more easily.

Real Solutions

Finding real solutions involves determining values that satisfy the original equation without any imaginary numbers. For a quadratic equation, these solutions are often the x-values where the function crosses the x-axis on a graph.To verify real solutions from a factored equation:

Set each factor equal to zero and solve for the variable.
Ensure these solutions satisfy the original equation when substituting back.

In the exercise, the factors $(u - 9)$ and $(u + 1)$ provided $ u = 9 $ and $ u = -1 $. Substituting $ u $ back as $ x+1 $ gave the solutions $ x = 8 $ and $ x = -2 $. Testing these in the original equation confirmed they are indeed real solutions.

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91影视

For the following exercises, solve the equation by identifying the quadratic form. Use a substitute variable and find all real solutions by factoring. $$ (x+1)^{2}-8(x+1)-9=0 $$

Short Answer

Step by step solution

Identify the Substitution Variable

Rewrite the Equation in Terms of the New Variable

Factor the Quadratic Equation

Solve for the Substitute Variable

Substitute Back to Original Variable

Verify the Real Solutions

Key Concepts

Factoring

Substitution Method

Real Solutions

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