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Chapter 1: Problem 64

Find all real values of \(x\) such that \(f(x)=0\). $$ f(x)=x^{2}-8 x+15 $$

Short Answer

Expert verified
The real values of \(x\) for which \(f(x) = 0\) are \(x = 5\) and \(x = 3\).

Step by step solution

01

Identify the coefficients

The quadratic equation is given as \(f(x)=x^{2}-8x+15\). So, here the coefficient of \(x^2\) which is \(a=1\), the coefficient of \(x\) which is \(b=-8\), and the constant term is \(c=15\).
02

Use the quadratic formula

We apply the quadratic formula which is \(-b 卤 \sqrt{b^2-4ac}/2a\). Substituting \(a\), \(b\), and \(c\) into the formula, we get \(x = \[8 卤 \sqrt{(-8)^2-4*1*15}/2*1\]\).
03

Simplify the equation

By simplifying the above equation, we obtain \(x = \[8 卤 \sqrt{64 - 60}\]/2 = 8 卤 \sqrt{4}/2\). So, \(x = \[8 卤 2\]/2\), which gives us two possible solutions, \(x = \[10/2 = 5\]\) or \(x = \[6/2 = 3\]\).

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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 set in the form of \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are coefficients, and \( a eq 0 \). Solving these equations means finding the values of \( x \) that make this equation true, also known as the roots. There are various methods to solve quadratic equations:
  • Factoring: This method is employed when the quadratic expression can be easily broken down into two binomial expressions. This method is often straightforward but might not always be applicable.
  • Completing the Square: This technique involves rearranging the equation and adding a constant term to make it a perfect square trinomial.
  • Quadratic Formula: This efficient method uses the general formula, which provides solutions for any quadratic equation, regardless of its factorability. We'll focus on this formula in detail.
Quadratic Formula
The quadratic formula is an essential tool in algebra and provides a systematic approach to finding the roots of any quadratic equation. It is given by:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
where \( a \), \( b \), and \( c \) come from the coefficients of the quadratic equation \( ax^2 + bx + c = 0 \). The formula consists of
  • Discriminant: Denoted by \( \sqrt{b^2 - 4ac} \), it determines the nature of the roots. If it's positive, there are two distinct real roots. If zero, one real root exists. If negative, the roots are complex.
  • Solution Calculation: Since the \( \pm \) symbol indicates two possibilities, the formula offers two solutions through addition and subtraction. By substituting values of \( a \), \( b \), and \( c \), solving gives:
    \( x = \frac{8 \pm \sqrt{4}}{2} \)
This approach ensures solutions even for challenging quadratic expressions.
Roots of Equations
The roots of an equation are the values of \( x \) that satisfy the equation \( ax^2 + bx + c = 0 \). They hold fundamental significance in understanding quadratic equations. In our example, using the quadratic formula, we determined:
  • Two Real Roots: For our equation \( f(x) = x^2 - 8x + 15 \), the roots obtained were \( x = 5 \) and \( x = 3 \). These represent the values where the graph of the equation intersects the x-axis.
  • Check Using Substitution: To validate, substitute \( x = 5 \) and \( x = 3 \) back into the original equation. If both satisfy the equation resulting in zero, the solutions are verified.
    \( f(5) = 5^2 - 8 \cdot 5 + 15 = 25 - 40 + 15 = 0 \)
    \( f(3) = 3^2 - 8 \cdot 3 + 15 = 9 - 24 + 15 = 0 \)
Finding roots provides a deeper understanding of the equation's behavior.

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Most popular questions from this chapter

Use the functions given by \(f(x)=\frac{1}{8} x-3\) and \(g(x)=x^{3}\) to find the indicated value or function. $$ g^{-1} \circ f^{-1} $$

Determine whether the function has an inverse function. If it does, find the inverse function. $$ f(x)=\left\\{\begin{array}{ll} -x, & x \leq 0 \\ x^{2}-3 x, & x>0 \end{array}\right. $$

Use the given value of \(k\) to complete the table for the inverse variation model $$y=\frac{k}{x^{2}}$$ Plot the points on a rectangular coordinate system. $$\begin{array}{|l|l|l|l|l|l|} \hline x & 2 & 4 & 6 & 8 & 10 \\ \hline y=\frac{k}{x^{2}} & & & & & \\ \hline \end{array}$$ $$ k=5 $$

Prove that if \(f\) is a one-to-one odd function, then \(f^{-1}\) is an odd function.

Use the functions given by \(f(x)=\frac{1}{8} x-3\) and \(g(x)=x^{3}\) to find the indicated value or function. $$ \left(g^{-1} \circ g^{-1}\right)(-4) $$
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