/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 44 Let \(F(x)=x^{2}+8 x+16 .\) Find... [FREE SOLUTION] | 91影视

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Chapter 11: Problem 44

Let \(F(x)=x^{2}+8 x+16 .\) Find \(x\) such that \(F(x)=9\).

Short Answer

Expert verified
The solutions for x are \(x = -1\) and \(x = -7\).

Step by step solution

01

- Set Up the Equation

We are provided with the function \(F(x)=x^2+8x+16\) and need to find the value of \(x\) for which \(F(x)=9\). Therefore, set the equation \(F(x)=9\) which gives us: \[x^2 + 8x + 16 = 9\].
02

- Move All Terms to One Side

Subtract 9 from both sides of the equation to set it to zero: \[x^2 + 8x + 16 - 9 = 0\]. Simplifying this, we get: \[x^2 + 8x + 7 = 0\].
03

- Factor the Quadratic Equation

Next, factor the quadratic equation \(x^2 + 8x + 7 = 0\). This can be written as \((x + 1)(x + 7) = 0\).
04

- Solve for x

Set each factor equal to zero and solve for \(x\): \[x + 1 = 0\] or \[x + 7 = 0\]. This gives the solutions \(x = -1\) and \(x = -7\).
05

鈥� Verify the Solutions

Substitute \(x = -1\) and \(x = -7\) back into the original function to ensure they satisfy \(F(x) = 9\). Both values will satisfy the equation as follows: \[-1^2 + 8(-1) + 16 = 9\] and \[-7^2 + 8(-7) + 16 = 9\].

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Key Concepts

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

Factoring Quadratic Equations
Factoring quadratic equations is an essential technique to solve for unknown variables in equations of the form \(ax^2 + bx + c = 0\). Factoring involves expressing the quadratic equation as a product of two binomials. These binomials will be set to zero and solved separately. Consider our given quadratic equation \(x^2 + 8x + 7\). To factor this, we look for two numbers that multiply to 7 (the constant term) and add up to 8 (the coefficient of the linear term). Here, the numbers are 1 and 7. Thus, we write:\( \[\begin{equation} x^2 + 8x + 7 = (x + 1)(x + 7) \end{equation}\] \) Now, we have factored the quadratic equation successfully!
Setting Equations to Zero
Before we can factor the quadratic equation, we need to set the equation to zero. This means that all terms must be on one side of the equation, leaving a zero on the other side. In the provided equation, we started with \(x^2 + 8x + 16 = 9\). To set this to zero, subtract 9 from both sides: \(x^2 + 8x + 16 - 9 = 0\). Simplified, this becomes \(x^2 + 8x + 7 = 0\). Setting the equation to zero is crucial because it allows us to apply factoring or other solution methods like the quadratic formula.
Verifying Solutions
After solving the quadratic equation by factoring, it's vital to verify the solutions to ensure they are correct. To do this, substitute each value of \(x\) back into the original equation. Let's verify the values \(x = -1\) and \(x = -7\). For \(x = -1\): \(((-1)^2 + 8(-1) + 16) = 1 - 8 + 16 = 9\). This confirms that \(x = -1\) is a solution. For \(x = -7\): \(((-7)^2 + 8(-7) + 16) = 49 - 56 + 16 = 9\). This confirms that \(x = -7\) is a solution. Verifying solutions is essential to ensure their validity in the context of the original equation.

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Most popular questions from this chapter

If \(f(x)=\left(x-\frac{1}{3}\right)\left(x^{2}+6\right)\) and \(g(x)=\) \(\left(x-\frac{1}{3}\right)\left(x^{2}-\frac{2}{3}\right),\) find all \(a\) for which \((f+g)(a)=0\).

Complete the square to find the \(x\) -intercepts of each function given by the equation listed. $$ f(x)=x^{2}+6 x+7 $$

For each equation under the given condition, (a) find \(k\) and (b) find the other solution. $$ k x^{2}-2 x+k=0 ; \text { one solution is }-3 $$

Complete the square to find the \(x\) -intercepts of each function given by the equation listed. $$ f(x)=x^{2}+10 x-2 $$

Show that whenever there is just one solution of \(a x^{2}+b x+c=0,\) that solution is of the form \(-b /(2 a)\).
See all solutions

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