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Chapter 9: Problem 25

Find the \(x\) -intercepts for the graph of each equation. $$ (x-5)^{2}-49=0 $$

Short Answer

Expert verified
The x-intercepts are \ x = 12 \ and \ x = -2 \.

Step by step solution

01

Rewrite the equation

The equation given is \( (x-5)^2 - 49 = 0 \). Rewrite it to make it easier to solve by moving \( -49 \) to the other side: \[ (x-5)^2 = 49 \]
02

Take the square root of both sides

To solve for \( x \), take the square root of both sides of the equation: \[ \sqrt{(x-5)^2} = \sqrt{49} \]. This simplifies to \| x-5 \| = 7.
03

Solve for x

Solve the absolute value equation \| x-5 \| = 7. This gives two solutions: \[ x-5 = 7 \] and \[ x-5 = -7 \]. Add 5 to both sides of each equation to find the x-values: \[ x = 12 \] and \[ x = -2 \]

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Key Concepts

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

solving quadratic equations
Quadratic equations are mathematical expressions that can be written in the standard form: \[ ax^2 + bx + c = 0 \] where \( a, b, \) and \( c \) are constants. Solving quadratic equations often involves finding the values of \( x \) that make the equation true, also known as the roots or solutions of the equation. There are several methods to solve these equations:
  • Factoring: Express the quadratic equation in a product of binomials.
  • Using the Quadratic Formula: \[ x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \]
  • Completing the square: Rewrite the equation in the form of a perfect square trinomial.
  • Graphing: Plot the equation on a graph and find the points where it intersects the x-axis.
In the exercise given, the equation \( (x-5)^2 - 49 = 0 \) was already in a form that allowed for easy manipulation. Moving the 49 to the other side and then taking the square root simplified the equation effectively.
absolute value
The absolute value of a number is its distance from zero on a number line, regardless of direction. The absolute value of \( x \) is written as \( |x| \). By definition:
  • \( |x| = x \) if \( x \geq 0 \)
  • \( |x| = -x \) if \( x < 0 \)
In the exercise, after taking the square root of both sides, we obtained the absolute value equation \( |x-5| = 7 \). This is because the square root of any non-negative number \( a^2 \) is \( |a| \). Solving \( |x-5| = 7 \) leads to two scenarios:
  • \( x-5 = 7 \)
  • \( x-5 = -7 \)
These solutions represent the points where the graph of the equation intersects the x-axis.
square roots
The square root of a number \( n \) is a value that, when multiplied by itself, gives \( n \). The square root is denoted by the symbol \( \sqrt{} \). For example, \( \sqrt{49} = 7 \) because \( 7 \times 7 = 49 \). When dealing with quadratic equations, taking the square root is often a crucial step. In our exercise, after isolating \( (x-5)^2 \), we took the square root of both sides, resulting in \( \sqrt{(x-5)^2} = \sqrt{49} \). This gave us \( |x-5| = 7 \), simplifying into two linear equations. It's important to remember that when you take the square root of both sides of an equation, you must consider both the positive and negative roots, as both will satisfy \( a^2 \).

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Most popular questions from this chapter

Find the \(x\) -intercepts for the graph of each equation. $$ 6 x^{2}+36-30 x=0 $$

Use the quadratic formula to solve each equation. $$ -6 a^{2}+3 a=-4 $$

Factor each expression. $$ 0.5 a^{2}-2 a-16 $$

Solve each inequality, and graph the solution on the number line. $$ 3 x-9<-4.5 x+6 $$

Suppose the \(2-\) of \(-6\) lottery game was modified so that after the first number was selected, that number was placed back into the group before the next number was selected. In this way, a number could be repeated, meaning pairs such as \(2-2\) and \(3-3\) would be possible. a. Would your chances of winning be better or worse for this modified game? Explain. b. How many possible pairs are there for this modified game, assuming that order does matter? Explain. c. List all of the possible pairs from part b. d. Since order really doesn't matter in this game, how many different pairs are there? (Remember, if order doesn't matter, \(1-2\) is the same as \(2-1 .\) ) e. Are all of the pairs considered in part d equally likely? Explain. f. If you choose one number pair for this modified game, what is the probability you will win. (Hint: There are two cases to consider.)
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