/*! 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 55 Solve each quadratic equation by... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 55

Solve each quadratic equation by factoring. $$x^{2}-3 x-10=0$$

Short Answer

Expert verified
The solutions to the quadratic equation \(x^2 - 3x - 10 = 0\) are \(x = 5\) and \(x = -2\).

Step by step solution

01

Write the equation in standard form

The quadratic equation is already in the standard form \(x^2 - 3x - 10 = 0\).
02

Factor the quadratic equation

To solve a quadratic equation by factoring, to begin with we look for two numbers that multiply to give -10 (the constant term) and add up to -3 (the coefficient of x). The numbers that satisfy this condition are -5 and 2. Therefore, we can rewrite the equation as follows: \( (x - 5)(x + 2) = 0 \).
03

Use the Zero Product Property to solve for x

The Zero Product Property tells us that if a product of factors equals zero, then at least one of the factors must be equal to zero. Therefore, we set each factor equal to zero and solve for x: \( x - 5 = 0 \) gives \( x = 5 \) and \( x + 2 = 0 \) gives \( x = -2 \).

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.

Factoring Quadratic Equations
When solving quadratic equations by factoring, the goal is to express the quadratic expression as a product of two binomials. To achieve this, one must find two numbers that both multiply to the constant term and add up to the coefficient of the linear term.

Let's consider the quadratic equation from the exercise \(x^2 - 3x - 10 = 0\). We need two numbers that multiply to \-10 and add to \-3. These numbers are \-5 and 2. We can then rewrite the equation by placing these two numbers into two binomials: \( (x - 5)(x + 2) = 0 \). The process commonly involves trial and error but following a systematic approach makes it achievable, especially with practice.
Zero Product Property
The Zero Product Property is a fundamental principle in algebra which states that if the product of several factors equals zero, at least one of the factors must also equal zero. This principle allows us to solve quadratic equations set in factored form, like \( (x - 5)(x + 2) = 0 \).

By applying the Zero Product Property, we set each factor in the equation equal to zero: \(x - 5 = 0\) and \(x + 2 = 0\). Solving these simple linear equations by adding 5 to the first and subtracting 2 from the second, gives us the solutions \(x = 5\) and \(x = -2\), respectively. It's an effective technique once a quadratic has been factored properly.
Standard Form of a Quadratic Equation
The standard form of a quadratic equation is expressed as \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are coefficients, and \(a\) is not equal to zero. The equation from our exercise, \(x^2 - 3x - 10 = 0\), is already in standard form where \(a = 1\), \(b = -3\), and \(c = -10\).

Recognizing the standard form is crucial as it provides a starting point for factoring and applying various methods such as completing the square or using the quadratic formula.
Roots of a Quadratic Equation
Roots of a quadratic equation, also known as solutions or x-intercepts, are the values of \(x\) for which the quadratic equation equals zero. In the context of our example \(x^2 - 3x - 10 = 0\), after factoring and applying the Zero Product Property, the roots were found to be \(x = 5\) and \(x = -2\).

Graphically, these roots represent the points at which the parabola, the curve representing the quadratic equation, crosses the x-axis. Determining the roots is essential for understanding the behavior of quadratic functions in calculus and other applications in mathematics.

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

Explain how to solve \(x^{2}+6 x+8=0\) using the quadratic formula.

Exercises \(159-161\) will help you prepare for the material covered in the next section. If 6.2 is multiplied by \(10^{3},\) what does this multiplication do to the decimal point in \(6.2 ?\)

Your local electronics store is having an end-of-the-year sale. The price on a plasma television had been reduced by \(30 \%\) Now the sale price is reduced by another \(30 \% .\) If \(x\) is the television's original price, the sale price can be modeled by $$(x-0.3 x)-0.3(x-0.3 x)$$ a. Factor out \((x-0.3 x)\) from each term. Then simplify the resulting expression. b. Use the simplified expression from part (a) to answer these questions. With a \(30 \%\) reduction followed by a \(30 \%\) reduction, is the television selling at \(40 \%\) of its original price? If not, at what percentage of the original price is it selling?

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The equation \(a x^{2}+c=0, a \neq 0,\) cannot be solved by the quadratic formula.

Write a quadratic equation in general form whose solution set is \(\\{-3,5\\}\)
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Mechanics Maths

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.