/*! 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 64 Solve the quadratic equation by ... [FREE SOLUTION] | 91影视

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Chapter 0: Problem 64

Solve the quadratic equation by factoring $$ -x^{2}+8 x=12 $$

Short Answer

Expert verified
The solutions of the quadratic equation \(-x^2 + 8x = 12\) are \(x = 6\) and \(x = 2\).

Step by step solution

01

Rearrange the equation

Rearrange the given equation into standard form. An equation can be considered in standard form when it equals 0. Therefore, \( -x^2 + 8x -12 = 0 \)
02

Factor the quadratic equation

In order to solve the equation we need to factor it. The factored form of the quadratic equation is \((x -6)(x -2) = 0\). When a product of two factors equals zero, then at least one of the factors must be zero.
03

Solve for \(x\)

To solve for \(x\), set each factor equal to zero and solve. \[ x - 6 = 0 \Rightarrow x = 6 \] and \[ x - 2 = 0 \Rightarrow x = 2 \]. So, the solutions to the equation are 6 and 2.

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Key Concepts

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

Factoring Quadratic Equations
Understanding how to factor quadratic equations is a fundamental skill in algebra. Factoring involves rewriting the equation as a product of its binomial factors. For instance, a quadratic equation in the form of \( ax^2 + bx + c = 0 \) can often be factored into \((x - p)(x - q) = 0\), where \(p\) and \(q\) are the roots of the equation.

To factor successfully, one must search for two numbers that not only multiply to give \(ac\) (the product of the coefficients of \(x^2\) and the constant term) but also add up to \(b\), the coefficient of \(x\). This step is pivotal as it breaks down the quadratic into solvable linear factors.
Quadratic Equations Standard Form
The standard form of a quadratic equation is given as \( ax^2 + bx + c = 0 \), where \(a\), \(b\), and \(c\) are coefficients and \(a eq 0\). The standard form is vital because it sets the stage for various methods of finding solutions, including factoring, completing the square, or using the quadratic formula.

To solve by factoring, the equation must first be rearranged in standard form where one side is equal to zero. In doing so, you are preparing the equation for the factorization process, as seen in the exercise where the equation \(-x^2 + 8x = 12\) is rearranged to \(-x^2 + 8x - 12 = 0\).
Solving for x
Solving for \(x\) is the ultimate goal when dealing with quadratic equations. After factoring the equation into binomials, as done in the step 2 of the provided exercise, you end up with potential solutions that are derived by setting each factor equal to zero.

This step is based on a fundamental principle in algebra that if the product of two factors is zero, at least one of them must be zero. By isolating \(x\) in the equations \(x - p = 0\) and \(x - q = 0\), you can find the values of \(x\) that satisfy the original quadratic equation.
Zero Product Property
The zero product property states that if the product of two numbers is zero, then at least one of the numbers must be zero. This property is a critical tool when solving quadratic equations by factoring.

Once a quadratic equation is factored into the form \((x - p)(x - q) = 0\), the zero product property allows us to conclude that either \(x - p = 0\) or \(x - q = 0\) or both. Each binomial set to zero gives us a potential solution. Consequently, the solutions to the factored equation are the values of \(x\) where each binomial is equal to zero.

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Most popular questions from this chapter

In Exercises 39-54, (a) find the inverse function of \(f\), (b) graph both \(f\) and \(f^{-1}\) on the same set of coordinate axes, (c) describe the relationship between the graphs of \(f\) and \(f^{-1}\), and (d) state the domain and range of \(f\) and \(f^{-1}\). $$ f(x)=x^{3}+1 $$

Your wage is \(\$ 8.00\) per hour plus \(\$ 0.75\) for each unit produced per hour. So, your hourly wage \(y\) in terms of the number of units produced is $$ y=8+0.75 x $$ (a) Find the inverse function. (b) What does each variable represent in the inverse function? (c) Determine the number of units produced when your hourly wage is \(\$ 22.25\).

Each function models the specified data for the years 1995 through 2005 , with \(t=5\) corresponding to 1995 . Estimate a reasonable scale for the vertical axis (e.g., hundreds, thousands, millions, etc.) of the graph and justify your answer. (There are many correct answers.) (a) \(f(t)\) represents the average salary of college professors. (b) \(f(t)\) represents the U.S. population. (c) \(f(t)\) represents the percent of the civilian work force that is unemployed.

Geometry A rectangle is bounded by the \(x\)-axis and the semicircle \(y=\sqrt{36-x^{2}}\) (see figure). Write the area \(A\) of the rectangle as a function of \(x\), and determine the domain of the function.

In Exercises 39-54, (a) find the inverse function of \(f\), (b) graph both \(f\) and \(f^{-1}\) on the same set of coordinate axes, (c) describe the relationship between the graphs of \(f\) and \(f^{-1}\), and (d) state the domain and range of \(f\) and \(f^{-1}\). $$ f(x)=\sqrt{4-x^{2}}, \quad 0 \leq x \leq 2 $$
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