/*! 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 42 Solve the quadratic equation by ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 42

Solve the quadratic equation by completing the square. Verify your answer graphically. $$-x^{2}+6 x-16=0$$

Short Answer

Expert verified
The roots of the quadratic equation \( -x^{2}+6x-16=0 \) are \( x = 3 + \sqrt{7}i \) and \( x = 3 - \sqrt{7}i \)

Step by step solution

01

Write the given equation in standard form

The given equation is already in standard form, i.e, \( -x^{2}+6x-16=0 \)
02

Move the constant term to the right side

To complete the square, move constant term on the other side of equation, giving \( -x^{2}+6x = 16 \)
03

Divide the equation by coefficient of x^2

Dividing each side by -1, we get \( x^2-6x = -16 \)
04

Move the coefficient of x to complete the square

Half the coefficient of \( x \) is -3. Square this value, i.e., (-3)虏 = 9, and add it to both sides, giving \( x^2-6x+9 = -16+9 \),which simplifies to \( x^2-6x+9 = -7 \)
05

Simplify the equation to find the roots

Taking square root on both sides, we get \( (x - 3) = \sqrt{-7} \). As there is a negative sign under the square root, there will be no real roots. Therefore, the roots are complex i.e. \( (x - 3) = \pm \sqrt{7}i \). So, the roots of the equation are \( x = 3 + \sqrt{7}i \) and \( x = 3 - \sqrt{7}i \)
06

Verify the roots graphically

Plot the function \( y = -x^{2}+6x-16 \). Since the roots are complex, they won't intersect the x-axis, which is true as per the graphical representation.

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.

Completing the Square
Completing the square is a method used to solve quadratic equations by turning a quadratic expression into a perfect square trinomial. This makes it easier to solve for the variable, especially when factoring isn't straightforward.

To complete the square for the quadratic equation \(-x^2 + 6x - 16 = 0\), we first need to rearrange it into the form \(x^2 - 6x = -16\) by moving the constant to the other side and ensuring the leading coefficient of \(x^2\) is positive.
  • Find half of the coefficient of \(x\), which is -3, and then square it to get 9.
  • Add 9 to both sides of the equation to maintain equality. This transforms the equation into \(x^2 - 6x + 9 = -7\).
Now, the left side of the equation is a perfect square trinomial, \((x - 3)^2\), which simplifies solving the equation further.
Complex Roots
When solving \((x - 3)^2 = -7\), taking the square root reveals a negative number under the radical. This indicates the presence of complex roots.

The expression becomes \(x - 3 = \pm \sqrt{7}i\). Here, \(i\) is the imaginary unit, defined as \(\sqrt{-1}\). Complex roots occur in conjugate pairs, that is, pairs like \(3 + \sqrt{7}i\) and \(3 - \sqrt{7}i\).
  • Complex roots mean the graph of the quadratic doesn't cross the x-axis.
  • This makes completing the square particularly useful, as it leads directly to the identification of complex solutions.
Graphical Verification
Graphically verifying the solution involves plotting the quadratic equation \(y = -x^2 + 6x - 16\). When you plot it:
  • The curve opens downwards, due to the negative sign in front of \(x^2\).
  • Since the roots are complex, the graph does not touch or intersect the x-axis.
Seeing that the parabola doesn't cross the x-axis confirms the absence of real roots and the presence of complex roots. This step ensures that the analytical solutions match the visual insight from the graph.
Standard Form of Quadratic
Quadratic equations are often expressed in standard form as \(ax^2 + bx + c = 0\). This format is crucial for applying various solution methods, like completing the square or the quadratic formula.

In our example, the equation \(-x^2 + 6x - 16 = 0\) is already in standard form. Here:
  • \(a = -1\), \(b = 6\), and \(c = -16\).
  • Maintaining this form simplifies the process of identifying necessary transformations or calculations.
The consistency of the standard form allows for systematic approaches to finding solutions, whether they are real or complex.

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

Solve the inequality and graph the solution on the real number line. Use a graphing utility to verify your solution graphically. $$\frac{x+6}{x+1}-2 \leq 0$$

Solve the inequality and graph the solution on the real number line. Use a graphing utility to verify your solution graphically. $$3 x^{2}-11 x+16 \leq 0$$

Solving an Equation Involving Fractions Find all solutions of the equation. Check your solutions. $$\frac{1}{x}-\frac{1}{x+1}=3$$

use the models below, which approximate the numbers of Bed Bath \(\&\) Beyond stores \(B\) and Williams-Sonoma stores \(W\) for the years 2000 through \(2013,\) where \(t\) is the year, with \(t=0\) corresponding to 2000. (Sources: Bed Bath \(\&\) Beyond, Inc.; Williams-Sonoma, Inc.) Bed Bath \& Beyond: $$B=86.5 t+342,0 \leq t \leq 13$$ Williams-Sonoma: $$W=-2.92 t^{2}+52.0 t+381,0 \leq t \leq 13$$. Solve the inequality \(B(t) \geq W(t) .\) Explain what the solution of the inequality represents.

Solve the inequality and graph the solution on the real number line. Use a graphing utility to verify your solution graphically. $$4 x^{2}+12 x+9 \leq 0$$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Geometry

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.