Problem 42 Review solving equations. Solv... [FREE SOLUTION] (2024)

Get started for free

Get started for free Log out

Chapter 13: Problem 42

Review solving equations. Solve $$ x^{2}+4 x=60[6.2] $$

Short Answer

Expert verified

The solutions are x = -10 and x = 6.

Step by step solution

Set the Equation to Zero

First, move all terms to one side of the equation to set it to zero. Subtract 60 from both sides: x^2 + 4x - 60 = 0

Factor the Quadratic Equation

Next, factor the quadratic equation. Find two numbers that multiply to -60 and add up to 4. These numbers are 10 and -6, hence the equation can be factored as: (x + 10)(x - 6) = 0

Set Each Factor to Zero and Solve

Now, set each factor equal to zero and solve for x: For x + 10 = 0x = -10. For x - 6 = 0x = 6.

Verify the Solutions

Finally, verify the solutions by plugging them back into the original equation:For x = -10, (-10)^2 + 4(-10)= 100 - 40 = 60which is true.For x = 6, (6)^2 + 4(6) = 36 + 24 = 60which is also true.

Key Concepts

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

setting equations to zero

To solve a quadratic equation, the first step is to set it to zero. This involves moving all terms to one side of the equation.
Doing this ensures we can use factoring or the quadratic formula later.
For our exercise $x^{2}+4x=60$, we begin by subtracting 60 from both sides to get:
\[ x^2 + 4x - 60 = 0 \]
This step is crucial because we need a zero on one side to factor the equation correctly.

factoring quadratic equations

Once the equation is set to zero, the next step is factoring the quadratic equation.
Factoring involves finding two numbers that multiply to the constant term (-60 in this example) and add up to the coefficient of the middle term (4 in this example).
For \[ x^2 + 4x - 60 = 0 \], we try to find two numbers that multiply to -60 and add up to 4.
These numbers are 10 and -6. So, we can write the equation as two binomials:
\[ (x + 10)(x - 6) = 0 \],
This factored form represents the equation of the quadratic function as two expressions.

verifying solutions

After factoring and solving for the variables, the final step is verifying the solutions.
This ensures that the solutions satisfy the original equation.
For our factored equation \[ (x + 10)(x - 6) = 0 \], solving for x gives us two solutions: $x = -10$ and $x = 6$.
To verify, substitute these values back into the original equation:

For $x = -10$: \[ (-10)^2 + 4(-10) = 100 - 40 = 60 \]
For $x = 6$: \[ (6)^2 + 4(6) = 36 + 24 = 60 \]

Both solutions satisfy the original equation, thus confirming their correctness.

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 ratio of the length to the height of the screen on a computer monitor is 4to $3 .$ A Dell Inspiron notebook has a $15-$ in. diagonal screen. Find thedimensions of the screen.Solve. $3 x^{2}+10 x-8=0$Find the center and the radius of each circle. Then graph the circle. $$x^{2}+y^{2}=36$$Find the center and the radius of each circle. Then graph the circle. $$ (x+1)^{2}+(y+3)^{2}=36 $$Match the equation with the center or vertex of its graph, listed in thecolumn on the right. a) Vertex: $(-2,5)$ b) Vertex: $(5,-2)$ c) Vertex: $(2,-5)$ d) Vertex: $(-5,2)$ e) Center: $(-2,5)$ f) Center: $(2,-5)$ g) Center: $(5,-2)$ h) Center: $(-5,2)$ $$y=(x-5)^{2}-2$$

See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

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.

Sign-up for free

This website uses cookies to improve your experience. We'll assume you're ok with this, but you can opt-out if you wish. Accept

Privacy & Cookies Policy

Privacy Overview

This website uses cookies to improve your experience while you navigate through the website. Out of these, the cookies that are categorized as necessary are stored on your browser as they are essential for the working of basic functionalities of the website. We also use third-party cookies that help us analyze and understand how you use this website. These cookies will be stored in your browser only with your consent. You also have the option to opt-out of these cookies. But opting out of some of these cookies may affect your browsing experience.

Necessary

Always Enabled

Necessary cookies are absolutely essential for the website to function properly. This category only includes cookies that ensures basic functionalities and security features of the website. These cookies do not store any personal information.

Non-necessary

Any cookies that may not be particularly necessary for the website to function and is used specifically to collect user personal data via analytics, ads, other embedded contents are termed as non-necessary cookies. It is mandatory to procure user consent prior to running these cookies on your website.

SAVE & ACCEPT