/*! 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 21 Use the quadratic formula to fin... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 21

Use the quadratic formula to find the zeros of the function \(f(x)=x^{2}-4 x+13\) and then write the function in factored form. Without graphing this function, how can you tell if it intersects the \(x\) -axis?

Short Answer

Expert verified
The zeros are \(2 \pm 3i\) and the function is written as \(f(x) = (x - (2 + 3i))(x - (2 - 3i))\). It doesn't intersect the x-axis because it has complex zeros.

Step by step solution

01

Identify the coefficients

For the quadratic function \(f(x) = x^2 - 4x + 13\), identify the coefficients: a = 1 (coefficient of x^2), b = -4 (coefficient of x), c = 13 (constant term).
02

Apply the quadratic formula

The quadratic formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Substitute the coefficients a, b, and c: \(x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(13)}}{2(1)} = \frac{4 \pm \sqrt{16 - 52}}{2}\).
03

Simplify the expression under the square root

Calculate the value under the square root: \(16 - 52 = -36\). So the quadratic formula becomes \(x = \frac{4 \pm \sqrt{-36}}{2}\).
04

Simplify the square root of a negative number

Since \(-36 = 36i^2\), we have \(\sqrt{-36} = 6i\). Substitute back into the equation: \(x = \frac{4 \pm 6i}{2}\).
05

Simplify the expression

Divide each term in the numerator by 2: \(x = 2 \pm 3i\). Thus, the solutions are \(x = 2 + 3i \) and \(x = 2 - 3i\).
06

Write the function in factored form

Using the solutions, \(f(x) = (x - (2 + 3i))(x - (2 - 3i))\).
07

Determine intersection with the x-axis

Since the solutions are complex \(2 + 3i \) and \(2 - 3i\), the function has no real zeros and does not intersect the x-axis.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

quadratic formula
The quadratic formula is a powerful tool for finding the zeros of any quadratic function. It gives the solutions of a quadratic equation of the form \(ax^2 + bx + c = 0\). The formula is:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
You just need to identify the coefficients \(a\), \(b\), and \(c\) from the equation. For the function \(f(x) = x^2 - 4x + 13\), we have:
\[ a = 1, \ b = -4, \ c = 13 \]
Plugging these values into the formula gives us:
\[ x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(13)}}{2(1)} \]
Simplifying, we get:
\[ x = \frac{4 \pm \sqrt{16 - 52}}{2} \]
Next, evaluate \-36\ under the square root.
complex numbers
When we have a negative number under the square root in the quadratic formula, our solutions involve complex numbers. For \-36\, we can rewrite it as \36i^2\, where \i\ is the imaginary unit with the property \i^2 = -1\. Thus, \[ \sqrt{-36} = 6i \]
Substituting back into the solution gives:
\[ x = \frac{4 \pm 6i}{2} \]
This simplifies to:
\[ x = 2 \pm 3i \]
The solutions, \2 + 3i\ and \2 - 3i\, are complex numbers.
factored form
We can express a quadratic function using its solutions. Since \ x = 2 + 3i \ and \ 2 - 3i \ are solutions, this means:
\[ (x - (2 + 3i))(x - (2 - 3i)) \]
can be used to write \ f(x) \ in its factored form. Therefore, the quadratic function becomes:
\[ f(x) = (x - (2 + 3i))(x - (2 - 3i)) \]
This representation showcases that it has complex roots.
x-axis intersection
To determine if a quadratic function intersects the \ x \-axis, we look at its solutions. If they are real, the function intersects the \ x \-axis, but if they are complex, it does not.
Since in our example \(f(x) = x^2 - 4x + 13\), the solutions \ 2 + 3i \ and \ 2 - 3i \ are complex, \ f(x) \ does not have any real solutions. Hence, the graph of \ f(x) \ does not intersect the x-axis.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A pilot has crashed in the Sahara Desert. She still has her maps and knows her position, but her radio is destroyed. Her only hope for rescue is to hike out to a highway that passes near her position. She needs to determine the closest point on the highway and how far away it is. a. The highway is a straight line passing through a point 15 miles due north of her and another point 20 miles due east. Draw a sketch of the situation on graph paper, placing the pilot at the origin and labeling the two points on the highway. b. Construct an equation that represents the highway (using \(x\) for miles east and \(y\) for miles north). c. Now use the Pythagorean Theorem to describe the square of the distance, \(d,\) of the pilot to any point \((x, y)\) on the highway. d. Substitute the expression for \(y\) from part (b) into the equation from part (c) in order to write \(d^{2}\) as a quadratic in \(x\) e. If we minimize \(d^{2}\), we minimize the distance \(d\). So let \(D=d^{2}\) and write \(D\) as a quadratic function in \(x\). Now find the minimum value for \(D\). f. What are the coordinates of the closest point on the highway, and what is the distance, \(d\), to that point?

Estimate the maximum number of horizontal intercepts for each of the polynomial functions. Then graph the function using technology to find the actual number. a. \(y=x^{4}-2 x^{2}-5\) b. \(y=4 t^{6}+t^{2}\) c. \(y=x^{3}-3 x^{2}+4\) d. \(y=5+x\)

In each part, construct a polynomial function with the indicated characteristics. a. Crosses the \(x\) -axis at least three times b. Crosses the \(x\) -axis at \(-1,3,\) and 10 c. Has a \(y\) -intercept of 4 and degree of 3 d. Has a \(y\) -intercept of -4 and degree of 5

The following tables represent a function \(f\) that converts cups to quarts and a function \(g\) that converts quarts to gallons (all measurements are for fluids). a. Fill in the missing values in the chart. (Hint: One quart contains 4 cups, and one gallon contains 4 quarts.) b. Now evaluate each of the following and identify the units of the results. i. \((g \circ f)(8)\) iii. \(\left(f^{-1} \circ g^{-1}\right)(1)\) ii. \(g^{-1}(2)\) iv. \(\left(f^{-1} \circ g^{-1}\right)(2)\) c. Explain the significance of \(\left(f^{-1} \circ g^{-1}\right)(x)\) in terms of cups, quarts, and gallons.

The winds are calm, allowing a forest fire to spread in a circular fashion at 5 feet per minute. a. Construct a function \(A(r)\) for the circular area burned, where \(r\) is the radius. Identify the units for the input and the output of \(A(r)\). b. Construct a function for the radius \(r=R(t)\) for the increase in the fire radius as a function of time \(t .\) What are the units now for the input and the output for \(R(t) ?\) c. Construct a composite function that gives the burnt area as a function of time. What are the units now for the input and the output? d. How much forest area is burned after 10 minutes? One hour?
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation

Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.