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Chapter 9: Problem 32

32\. \(7 x^{2}=84\)

Short Answer

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

Step by step solution

01

- Isolate the variable term

The given equation is \(7x^{2}=84\). To isolate the term with the variable, divide both sides of the equation by 7.
02

- Simplify the equation

After dividing both sides by 7, the equation simplifies to \(x^{2}=12\).
03

- Solve for the variable

To solve for \(x\), take the square root of both sides of the equation. This gives \(x = \pm \sqrt{12}\).
04

- Simplify the square root

Simplify \(\sqrt{12}\) to \(2\sqrt{3}\). Thus, the solutions are \(x = 2\sqrt{3}\) and \(x = -2\sqrt{3}\).

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Key Concepts

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

Isolate the Variable
In solving quadratic equations, the first step is often to isolate the variable term. This means you want to get the term with the variable on one side of the equation by itself.

For example, in the equation \(7x^{2}=84\), to isolate the variable term \(7x^{2}\), we need to get rid of the number 7 that is multiplying \(x^{2}\).

We do this by dividing both sides of the equation by 7, which simplifies the equation. After dividing by 7, the equation becomes \(x^{2}=12\).

This step is crucial as it sets us up for the next stages of solving the equation.
Simplifying Equations
Simplifying an equation means to make it as straightforward as possible. After isolating the variable term, you often need to perform operations that make the equation easier to solve.

In our example, once we divided both sides by 7 and got \(x^{2}=12\), the equation is much simpler.

By simplifying, we prepare the equation for easier manipulation in the later steps. A simplified equation reduces the chances of making errors and helps in understanding the equation more clearly.
Square Roots
Taking the square root is a common step in solving quadratic equations. Once the equation is simplified, you often need to find the value of the variable by taking the square root of both sides.

In the simplified equation \(x^{2}=12\), taking the square root of both sides helps to find \(x\) quickly. When you take the square root of both sides, it results in two values: one positive and one negative.

Therefore, \(x = \pm \sqrt{12}\). This is because both \( (2\sqrt{3})^{2} \) and \( (-2\sqrt{3})^{2} \) equal 12.
Quadratic Solutions
Finding the solutions to a quadratic equation usually involves multiple steps that we've discussed, including isolating the variable, simplifying the equation, and taking the square root.

Once you take the square root and find \(x = \pm \sqrt{12}\), it's important to simplify the square root itself. Simplifying \(\sqrt{12}\) gives \(\sqrt{4 \times 3} = 2\sqrt{3}\).

Therefore, the final solutions to the equation \(7x^{2}=84\) are \(x = 2\sqrt{3}\) and \(x = -2\sqrt{3}\). These are your quadratic solutions.

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Most popular questions from this chapter

Problem: The vertex of a parabola that opens up is \((4,-5)\). Write the equation of the axis of symmetry. Incorrect

In the formula \(C=L+S(D+H), C\) is the number of linear inches of carpet needed to cover a staircase and landing, the number of steps on the stairs is \(S\), the depth of each step in inches is \(D\), the height of each step in inches is \(H\), and the length of the landing in inches is \(L\). A staircase has 10 steps. Each step is \(7.25\) in. deep and \(10.25\) in. high. There is one 3 -ft landing. Find the length of carpet in yards needed to cover this staircase and landing. ( \(1 \mathrm{yd}=3 \mathrm{ft} ; 1 \mathrm{ft}=12\) in.) Round up to the nearest whole number.

\(y=-x^{2}+3\)

\(v^{2}+2 v+8=0\)

\(m^{2}+24 m+167=0\)
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