/*! 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 9 Solve the quadratic equation by ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 9

Solve the quadratic equation by factoring. $$x^{2}-25=0$$

Short Answer

Expert verified
The solution to the quadratic equation \(x^{2}-25=0\) is \(x=5\) and \(x=-5\).

Step by step solution

01

Write the Equation

The given equation is \(x^{2} - 25 = 0\).
02

Factor the Equation

The equation is a difference of squares, and can be factored into \((x-5)(x+5)=0\)
03

Use Zero Product Property

According to the zero product property, if a*b = 0, then a = 0 or b = 0. Hence, set each factor equal to zero and solve for x. This gives \(x-5=0\) and \(x+5=0\), leading to the solutions \(x=5\) and \(x=-5\).

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
Factoring is a method used to simplify an equation or expression into simpler terms that, when multiplied together, return to the original equation. In the context of quadratic equations, factoring is often used to break down a complex equation into the product of two binomials.

Here鈥檚 a quick step-by-step guide to factoring quadratics:
  • Identify a quadratic equation in the form of \(ax^2 + bx + c = 0\).
  • Look for two numbers that multiply to \(ac\) and add to \(b\).
  • Rearrange the original equation using the two numbers found.
  • Factor by grouping if needed, or simply write as two binomials.
This method helps in solving quadratic equations by simplifying them to basic forms like \((x-m)(x-n) = 0\),which leads directly into finding possible values of \(x\). If you remember these easy steps, factoring becomes an approachable and handy tool for solving quadratics.
Difference of Squares
The difference of squares is a special case in algebra, specifically with quadratic equations, where an expression is of the form \(a^2 - b^2\). This unique form can be factored into two conjugate binomials: \((a-b)(a+b)\).

Recognizing the difference of squares can simplify solving quadratics immensely:
  • Ensure the expression fits the pattern \(x^2 - c^2\).
  • Write the equation as the product of the binomials \((x-c)(x+c)\).
Example: In the equation \(x^2 - 25 = 0\),we identify\(25 = 5^2\). Hence, it's a difference of squares, allowing us to factor it easily as \((x-5)(x+5) = 0\).

This pattern is a common shortcut in algebra that reduces time and effort when solving quadratic equations.
Zero Product Property
The zero product property is an essential concept in algebra that states if the product of two numbers equals zero, then at least one of the numbers must be zero. Mathematically, if \(a \cdot b = 0\),then \(a = 0\)or\(b = 0\).

Using this property, solving factored quadratic equations like \((x-5)(x+5) = 0\)becomes straightforward:
  • Set each factor equal to zero: \(x-5 = 0\) and \(x+5 = 0\).
  • Solve both equations separately to find the values of \(x\): \(x = 5\)and\(x = -5\).
This principle makes it easy to determine the solutions to quadratic equations once factored, providing a clear and efficient pathway to finding the roots of the equation.

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

The production cost, in dollars, for \(x\) color brochures is \(C(x)=500+3 x .\) The fixed cost is \(\$ 500\) since that is the amount of money needed to start production even if no brochures are printed. (a) If the fixed cost is decreased by \(\$ 50,\) find the new cost function. (b) Graph both cost functions and interpret the effect of the decreased fixed cost.

Can you write down an expression for a quadratic function whose \(x\) -intercepts are given by (2,0) and (3,0)\(?\) Is there more than one possible answer? Explain.

The point (-2,2) on the graph of \(f(x)=|x|\) has been shifted horizontally and vertically to the point (3,4) Identify the shifts and write a new function \(g(x)\) in terms of \(f(x)\).

The perimeter of a square is \(P(s)=4 s\) where \(s\) is the length of a side in inches. The function \(C(x)=2.54 x\) takes \(x\) inches as input and outputs the equivalent result in centimeters. Find \((C \circ P)(s)\) and explain what it represents.

The Washington Redskins' revenue can be modeled by the function \(R(t)=245+40 t,\) where \(t\) is the number of years since 2003 and \(R(t)\) is in millions of dollars. The team's operating costs are modeled by the function \(C(t)=170+60 t,\) where \(t\) is the number of years since 2003 and \(C(t)\) is in millions of dollars. Find the profit function \(P(t) .\) (Source: Associated Press)
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

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.