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Chapter 8: Problem 36

See Example 5. Let \(f(x)=(x+3)^{2} .\) For what value(s) of \(x\) is \(f(x)=7 ?\)

Short Answer

Expert verified
The values of \(x\) are \(\sqrt{7} - 3\) and \(-\sqrt{7} - 3\).

Step by step solution

01

Set the equation

We start by setting up the equation based on the problem statement. We have the function \[ f(x) = (x + 3)^2. \] We need to find the value(s) of \(x\) for which \(f(x) = 7\). Therefore, we set the equation:\[ (x + 3)^2 = 7. \] Our goal is to solve this equation for \(x\).
02

Solve for the Square Root

We need to eliminate the square by taking the square root of both sides of the equation. This will give us two possible values since the square root has both positive and negative roots:\[ x + 3 = \pm \sqrt{7}. \]
03

Isolate x

Next, solve each of the two equations for \(x\):For the positive root:\[ x + 3 = \sqrt{7} \ x = \sqrt{7} - 3. \]For the negative root:\[ x + 3 = -\sqrt{7} \ x = -\sqrt{7} - 3. \]
04

Finalize the solution

The solutions we obtained from isolating \(x\) are:\[x = \sqrt{7} - 3\]and\[x = -\sqrt{7} - 3.\]These are the values of \(x\) for which \(f(x) = 7\).

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Key Concepts

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

Quadratic Equations
Quadratic equations are fundamental in algebra and are typically expressed in the standard form: \( ax^2 + bx + c = 0 \). Here, \( a \), \( b \), and \( c \) are constants, with \( a eq 0 \). In the given exercise, the quadratic expression is part of a function \( f(x) = (x+3)^2 \), which doesn't require rearranging into the standard form to find its roots. Instead, we set it equal to a specific value and solve.
  • The essence of quadratic equations lies in their ability to model parabolas in graphs, representing various real-world scenarios like projectile motion.
  • Quadratic equations can have two real solutions, one real solution, or complex solutions depending on the discriminant value \( b^2 - 4ac \).
Understanding quadratics involves recognizing how they behave when graphed and how to manipulate equations algebraically to find the roots.
Square Roots
The square root operation is a mathematical function that, for a given number \( y \), returns a number \( x \) such that \( x^2 = y \). In our problem, square roots are used to remove the power of 2 in the equation \((x + 3)^2 = 7\).
  • When taking square roots, we obtain two values: a positive and a negative (\( \sqrt{y} \) and \( -\sqrt{y} \)) because both square to give \( y \).
  • Square roots are essential in simplifying quadratic equations to extract solutions.
It is important to remember that square roots have both a principal (positive) and secondary (negative) root, which is why we consider both during the solving process.
Solving Equations
Solving equations involves finding the unknown values that satisfy the equation. The approach depends on the type of equation. In our exercise, we solved a quadratic equation using the square root method:
  • The primary step in solving is to isolate the key term involving the variable. Here, it was \((x+3)^2\).
  • Next, apply basic mathematical operations like taking the square root to simplify the equation into solvable parts.
  • Finally, isolate \( x \) to determine its value by moving other terms across the equation using operations such as addition or subtraction.
These steps help in systematically reducing complex equations to find viable solutions for the unknowns.

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Most popular questions from this chapter

Solve each equation. Approximate the solutions to the nearest hundredth when appropriate. $$x^{2}+5 x-5=0$$

Dances. Tickets to a school dance cost $$ 4\( and the projected attendance is 300 people. It is further projected that for every 10 e increase in ticket price, the average attendance will decrease by \)5 .\( At what ticket price will the receipts from the dance be \)\$ 1,248 ?$

a. \(x^{2}-42 x+441=0\) b. \(x^{2}+42 x+441=0\)

Use a graphing calculator to solve each equation. If an answer is not exact, round to the nearest hundredth. See Using Your Calculator: Solving Quadratic Equations Graphically. $$ 0.5 x^{2}-0.7 x-3=0 $$

Determine a quadratic function whose graph has \(x\) -intercepts of \((2,0)\) and \((-4,0)\)
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