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Chapter 6: Problem 55

$$ \text { For Problems 45-68, solve each equation. (Objective 2) } $$ $$ 3 x^{2}=75 $$

Short Answer

Expert verified
The solutions are \( x = 5 \) and \( x = -5 \).

Step by step solution

01

Understand the Equation

We start with the equation \( 3x^2 = 75 \). Our goal is to solve for \( x \) by isolating \( x \) on one side of the equation.
02

Divide by 3

To isolate \( x^2 \), divide both sides of the equation by 3: \[ \frac{3x^2}{3} = \frac{75}{3} \] which simplifies to \( x^2 = 25 \).
03

Take the Square Root

Now, take the square root of both sides of the equation to solve for \( x \): \[ x = \pm \sqrt{25} \].
04

Simplify the Square Root

Calculate the square root: \( \sqrt{25} = 5 \). Therefore, \( x = \pm 5 \).
05

State the Solution

The solutions to the equation \( 3x^2 = 75 \) are \( x = 5 \) and \( x = -5 \).

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Key Concepts

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

Solving Equations
When solving equations, you aim to find the values of the unknown variable that satisfy the given equation. This involves several logical steps.
Let's walk through the process with the quadratic equation from the exercise: \( 3x^2 = 75 \).
  • First, identify what the equation is asking. Here, you're looking for the values of \( x \) that make the equation true.
  • Next, perform operations to isolate \( x \) on one side of the equation. This often involves multiple algebraic manipulations.

Start by dividing both sides by the coefficient of \( x^2 \) (which is 3 in this case), simplifying to \( x^2 = 25 \).
  • Check your work at each step to ensure accuracy. Small mistakes can lead to incorrect solutions.

Once you have a manageable expression, you're ready to tackle further steps using algebraic tools like square roots.
Square Roots
The square root operation is a common mathematical tool, especially important in quadratic equations. It involves identifying a value that, when multiplied by itself, gives the original number.
For the equation \( x^2 = 25 \), taking the square root of both sides helps simplify it, allowing you to isolate \( x \).
  • Apply the square root to both sides: \( x = \pm \sqrt{25} \).
  • The "\( \pm \)" symbol indicates there are two possible solutions: one positive and one negative.

Calculating further, \( \sqrt{25} = 5 \). Thus, \( x \) can be either 5 or -5.
This is because both \( 5^2 \) and \( (-5)^2 \) equal 25.
If you understand square roots, solving equations becomes more intuitive.
Algebraic Manipulation
Algebraic manipulation involves rearranging and simplifying equations to solve for a variable. This is crucial in managing complex expressions. Here, we apply several manipulations:
  • Dividing both sides of the equation \( 3x^2 = 75 \) by 3 simplifies the problem considerably.
  • Next, taking the square root of \( x^2 = 25 \) isolates our variable \( x \), revealing potential solutions.

Throughout these processes, it's vital to keep equations balanced by performing the same operations on each side. This maintains equality and leads you to the correct solution.
Algebraic manipulation can initially seem challenging, but practice helps build a strong understanding of how to tackle similar algebra problems methodically.

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Most popular questions from this chapter

$$ \text { For Problems } 41-70 \text {, solve each equation. (Objective } 3 \text { ) } $$ $$ (2 n+5)(n+4)=-1 $$

$$ \text { For Problems } 41-70 \text {, solve each equation. (Objective } 3 \text { ) } $$ $$ x^{3}+10 x^{2}+25 x=0 $$

$$ \text { For Problems } 41-70 \text {, solve each equation. (Objective } 3 \text { ) } $$ $$ (3 n-1)(4 n-3)=0 $$

\(x^{2}-18 x+45=0\)

$$ \text { For Problems } 41-70 \text {, solve each equation. (Objective } 3 \text { ) } $$ $$ -3 x^{2}-24 x=0 $$
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