/*! 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 108 Solve each quadratic equation by... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 108

Solve each quadratic equation by the method of your choice. $$x^{2}=6 x-7$$

Short Answer

Expert verified
The solutions for the given quadratic equation \(x^{2}=6 x-7\) are \(x = 3 + \sqrt{2}\) and \(x = 3 - \sqrt{2}\).

Step by step solution

01

Rearrange the equation

First, the given quadratic equation, \(x^{2}=6 x-7\), must be rearranged to the standard form, where everything is set equal to zero on one side of the equation. This can be achieved by subtracting \(6x\) and adding \(7\) to both sides, which gives the rearranged equation: \(x^2 - 6x + 7 = 0\)
02

Apply the quadratic formula

The next step is solving for the variable x, which can be accomplished through the quadratic formula \(x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}\), where a, b, and c are coefficients in the quadratic equation \(ax^2 + bx + c = 0\). For our rearranged equation \(x^2 - 6x + 7 = 0\), a=1, b=-6, and c=7. Substituting these values into the quadratic formula gives \(x=\frac{6 \pm \sqrt{(-6)^2-4*1*7}}{2*1}\)
03

Simplify the solution

Now, simplify the equation as follows: \(x=\frac{6 \pm \sqrt{36-28}}{2}\), which results in \(x=\frac{6 \pm \sqrt{8}}{2}\), it simplifies further into \(x = 3 \pm \sqrt{2}\)

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
Understanding how to solve quadratic equations is a fundamental skill in algebra. One reliable method that works for all types of quadratic equations is the quadratic formula. This formula directly calculates the roots of any quadratic equation in the form of \( ax^2 + bx + c = 0 \).

The quadratic formula is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \(a\), \(b\), and \(c\) are coefficients from the equation. The symbol \(\pm\) indicates that there will typically be two solutions: one with a positive square root and one with a negative. This is due to any squared number having both a positive and a negative root.

Understanding this concept is crucial as it plays a fundamental role in various applications, ranging from physics to engineering fields. Always remember to check if you can simplify the square root for a more exact answer.
Rearranging Equations
Rearranging equations is an essential preliminary step before applying the quadratic formula. Any quadratic equation must be rearranged into standard form \(ax^2 + bx + c = 0\) to apply the formula correctly. This involves combining like terms and moving all parts of the equation to one side, so the opposite side equals zero.

In the given exercise \(x^2 = 6x - 7\), rearranging means subtracting \(6x\) and adding \(7\) from both sides to get \(x^2 - 6x + 7 = 0\). By performing these steps, you're ensuring that the equation is set up properly for the application of the quadratic formula, increasing your chances of finding the correct solutions.
Standard Form Quadratic
The standard form of a quadratic equation is its most recognizable version: \( ax^2 + bx + c = 0 \), where \(a\), \(b\), and \(c\) are coefficients, and \(a \eq 0\). The reason \(a\) cannot be zero is that it would eliminate the squared term, thus turning the equation into a linear one.

Recognizing a standard form quadratic quickly proves useful when determining the most appropriate method for solving it. Once you achieve this form, it's evident that either factoring, completing the square, or using the quadratic formula, as in our exercise, is the next logical step. Each term \(a\), \(b\), and \(c\) plays a significant role and directly influences the shape and position of the parabola on a graph if the function \(y = ax^2 + bx + c\) is plotted.

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

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. A city commission has proposed two tax bills. The first bill requires that a homeowner pay \(\$ 1800\) plus \(3 \%\) of the assessed home value in taxes. The second bill requires taxes of \(\$ 200\) plus \(8 \%\) of the assessed home value. What price range of home assessment would make the first bill a better deal?

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. A company manufactures and sells blank audiocassette tapes. The weekly fixed cost is \(\$ 10,000\) and it costs \(\$ 0.40\) to produce each tape. The selling price is \(\$ 2.00\) per tape. How many tapes must be produced and sold each week for the company to generate a profit?

Each group member should research one situation that provides two different pricing options. These can involve areas such as public transportation options (with or without discount passes), cellphone plans, long-distance telephone plans, or anything of interest. Be sure to bring in all the details for each option. At a second group meeting, select the two pricing situations that are most interesting and relevant. Using each situation, write a word problem about selecting the better of the two options. The word problem should be one that can be solved using a linear inequality. The group should turn in the two problems and their solutions.

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. To earn an A in a course, you must have a final average of at least \(90 \% .\) On the first four examinations, you have grades of \(86 \%, 88 \%, 92 \%,\) and \(84 \% .\) If the final examination counts as two grades, what must you get on the final to earn an A in the course?

Explain how to add rational expressions having no common factors in their denominators. Use \(\frac{3}{x+5}+\frac{7}{x+2}\) in your explanation.
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

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.