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Chapter 11: Problem 69

Complete the square to find the \(x\) -intercepts of each function given by the equation listed. $$ f(x)=x^{2}-10 x-22 $$

Short Answer

Expert verified
The x-intercepts are \( x = 5 + \sqrt{47} \) and \( x = 5 - \sqrt{47} \).

Step by step solution

01

Rewrite the equation

Start by rewriting the equation in the form \( f(x) = x^2 - 10x - 22 \).
02

Move the constant term

Move the constant term \(-22\) to the right side of the equation: \( x^2 - 10x = 22 \).
03

Find the value to complete the square

To complete the square, take half of the coefficient of \( x \) (which is -10), square it, and add it to both sides of the equation:\( \left( \frac{-10}{2} \right)^2 = 25 \). Therefore, add 25 to both sides:\( x^2 - 10x + 25 = 22 + 25 \).
04

Simplify and write as a square

Simplify the right side and write the left side as a square:\( (x-5)^2 = 47 \).
05

Solve for x

Take the square root of both sides of the equation:\( x - 5 = \pm \sqrt{47} \).So, \( x = 5 + \sqrt{47} \) or \( x = 5 - \sqrt{47} \).
06

Find the x-intercepts

Since the x-intercepts are the values of \( x \) when \( f(x) = 0 \), the x-intercepts are:\( x = 5 + \sqrt{47} \) and \( x = 5 - \sqrt{47} \).

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Key Concepts

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

Quadratic Equations
A quadratic equation is a second-order polynomial of the form: \( ax^2 + bx + c = 0 \) where
  • \( a \) is the coefficient of the quadratic term \( x^2 \)

  • \( b \) is the coefficient of the linear term \( x \)

  • \( c \) is the constant term.
Quadratic equations graph as parabolas and can have one, two, or no real solutions. Real solutions of a quadratic equation are the x-intercepts where the graph of the equation crosses the x-axis. To solve quadratic equations, we can use various methods such as factoring, completing the square, and the quadratic formula.
X-Intercepts
X-intercepts are the points where the graph of a function touches or crosses the x-axis. In other words, they are the points where \( y = 0 \). For the quadratic function \( f(x) \), the x-intercepts correspond to the solutions of the equation: \( f(x) = 0 \). Finding x-intercepts is essential because they give us valuable information about the behavior and roots of the function. In our exercise, we complete the square to find the x-intercepts for the function \( f(x) = x^2 - 10x - 22 \). By solving \( (x-5)^2 = 47 \), we determine the x-intercepts as \( x = 5 + \sqrt{47} \) and \( x = 5 - \sqrt{47} \).
Square Roots
Square roots are mathematical operations that, when applied to a number, yield a value which, when multiplied by itself, gives the original number. In symbols: \( \sqrt{n} * \sqrt{n} = n \) This concept is vital in solving quadratic equations, especially when we complete the square. The general equation when we complete the square is: \( (x 鈥� h)^2 = k \). Taking the square root of both sides of this equation, we get two possible values: \( x - h = \sqrt{k} \) or \( x - h = - \sqrt{k} \). In our exercise, after completing the square and setting it equal to 47, we took the square root of both sides: \( \sqrt{(x-5)^2} = 卤 \sqrt{47} \),and found that \( x = 5 卤 \sqrt{47} \).
Factoring
Factoring involves breaking down a complex expression into simpler terms that, when multiplied, give the original expression. For quadratic equations, factoring is one method to find roots or x-intercepts. However, not all quadratic equations can be factored easily. For example, the quadratic equation \( x^2 - 10x - 22 \)is not easily factorable with simple integer factors. In such cases, alternative methods like completing the square or using the quadratic formula are more effective. While teaching factoring, ensure to cover:
  • Recognizing factorable quadratic equations
  • Standard forms for factoring, like \( (x + p)(x + q) = 0 \)
  • Special cases e.g., difference of squares.

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Most popular questions from this chapter

Complete the square to find the \(x\) -intercepts of each function given by the equation listed. $$ f(x)=x^{2}-8 x-10 $$

Solve. $$ 4 d^{2}+81=0 $$

Solve by completing the square. Show your work. $$ t^{2}-10 t=-23 $$

Solve. $$ 3 t^{2}-1=6 $$

If the graphs of \(f(x)=a_{1}\left(x-h_{1}\right)^{2}+k_{1}\) and \(g(x)=a_{2}\left(x-h_{2}\right)^{2}+k_{2}\) have the same shape, what, if anything, can you conclude about the \(a^{\prime}\) s. the \(h^{\prime} s,\) and the \(k^{\prime} s ?\) Why?
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