/*! 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 42 Use the square root property to ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 42

Use the square root property to solve each equation. \((5 t+2)^{2}=12\)

Short Answer

Expert verified
The solutions are \( t = \frac{2\sqrt{3} - 2}{5} \) and \( t = \frac{-2\sqrt{3} - 2}{5} \).

Step by step solution

01

- Isolate the squared term

The equation given is \( (5t + 2)^2 = 12 \). We can already see that the squared term \( (5t + 2)^2 \) is isolated on one side of the equation.
02

- Apply the square root property

Take the square root of both sides of the equation to remove the exponent. This gives us: \[ \sqrt{(5t + 2)^2} = \pm \sqrt{12} \]. Simplifying both sides, we get: \[ 5t + 2 = \pm \sqrt{12} \].
03

- Simplify the square root

Simplify the square root term: \[ \sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3} \]. So the equation becomes: \[ 5t + 2 = \pm 2\sqrt{3} \].
04

- Solve for t

We now have two separate equations to solve: \[ 5t + 2 = 2\sqrt{3} \] and \[ 5t + 2 = -2\sqrt{3} \]. Solve each equation for \( t \): \[ 5t = 2\sqrt{3} - 2 \] so \( t = \frac{2\sqrt{3} - 2}{5} \) and \[ 5t = -2\sqrt{3} - 2 \] so \( t = \frac{-2\sqrt{3} - 2}{5} \).

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.

solving quadratic equations
Quadratic equations are equations of the form ax^2 + bx + c = 0, where a, b, and c are constants, and x is the variable. The main goal when solving quadratic equations is to find the values of x that make the equation true. There are various methods to solve quadratic equations, such as factoring, completing the square, using the quadratic formula, and applying the square root property.

The square root property is particularly useful for equations that can be written in the form (x - p)^2 = q. Here's a step-by-step to help understand using the square root property:
  • First, isolate the squared term on one side of the equation.
  • Next, take the square root of both sides of the equation. Remember to consider both the positive and the negative square roots.
  • Finally, solve the resulting simple equations.
This method allows us to quickly solve many quadratic equations, especially those that are perfectly set up for the square root property.
isolating the squared term
Isolating the squared term is a key step when solving quadratic equations using the square root property. By having the squared term on one side of the equation and everything else on the other, it becomes much simpler to take the square root of both sides.

Let's look at an example: (5t + 2)^2 = 12. Here, the squared term is (5t + 2)^2. Notice that it is already isolated on one side of the equation. In many other problems, you may need to manipulate the equation a little bit to isolate the squared term.

For example, consider (t + 3)^2 - 5 = 20. You would first add 5 to both sides to get: (t + 3)^2 = 25. Now the squared term (t + 3)^2 is isolated, and you can proceed to use the square root property.
simplifying radicals
Simplifying radicals is an essential step in solving many quadratic equations. Radicals, often called roots, need to be in their simplest form to make final solutions more understandable.

Take the equation: 5t + 2 = 卤 鈭�12. To simplify, you would break down 鈭�12 into 鈭�4 and 鈭�3.

Since 4 is a perfect square ( 2^2), 鈭�4 = 2. Therefore, 鈭�12 simplifies to:

  • 鈭�12 = 鈭�(4 x 3) = 2鈭�3.
This makes it much easier to handle in equations. Continuing, the equation 5t + 2 = 卤2鈭�3 is more manageable.

In general, simplifying radicals helps to make both the steps and the final solutions clearer and simpler to understand.

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

Solve each equation. Check the solutions. $$-x=\sqrt{\frac{8-2 x}{3}}$$

Solve each inequality, and graph the solution set. $$ (x+2)(4 x-3)(2 x+7) \geq 0 $$

Solve each problem. When appropriate, round answers to the nearest tenth. An object is projected directly upward from the ground. After \(t\) seconds its distance in feet above the ground is $$ s(t)=144 t-16 t^{2} $$ After how many seconds will the object be \(128 \mathrm{ft}\) above the ground? (Hint: Look for a common factor before solving the equation.)

Solve each inequality, and graph the solution set. $$ \frac{2 x-3}{x^{2}+1} \geq 0 $$

A washing machine can be filled in \(6 \mathrm{~min}\) if both the hot water and the cold water taps are fully opened. Filling the washer with hot water alone takes 9 min longer than filling it with cold water alone. How long does it take to fill the washer with cold water?
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Geometry

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.