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Chapter 9: Problem 15

Solve each equation. $$\frac{4}{3} x^{2}+2=110$$

Short Answer

Expert verified
The solutions are \( x = 9 \) and \( x = -9 \).

Step by step solution

01

Subtract 2 from both sides

To isolate the term with the variable, first subtract 2 from both sides of the equation:\[ \frac{4}{3}x^2 + 2 - 2 = 110 - 2 \]This simplifies to:\[ \frac{4}{3}x^2 = 108 \]
02

Multiply both sides by the reciprocal of \(\frac{4}{3}\)

The reciprocal of \(\frac{4}{3}\) is \(\frac{3}{4}\). Multiply both sides of the equation by \(\frac{3}{4}\):\[ \left( \frac{3}{4} \cdot \frac{4}{3} \right)x^2 = 108 \cdot \frac{3}{4} \]This simplifies to:\[ x^2 = 81 \]
03

Take the square root of both sides

To solve for \(x\), take the square root of both sides:\[ \sqrt{x^2} = \sqrt{81} \]This simplifies to:\[ x = \pm 9 \]

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Key Concepts

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

Solving Quadratic Equations
A quadratic equation is an equation in which the highest exponent of the variable is 2. In general, it takes the form: \(ax^2 + bx + c = 0\). Here, 'a', 'b', and 'c' are constants. Let's align this with our presented problem, which has a quadratic term in the form of \(x^2\)
In this exercise, we are tasked with solving the equation \(\frac{4}{3} x^{2}+2=110\).
To solve a quadratic equation, follow these steps:
  • First, align all terms involving \(x^2\) on one side of the equation, and constants on the other.
  • Next, isolate the \(x^2\) term by performing operations such as addition or subtraction.
  • Finally, take the square root of both sides to resolve the value of the variable.
In essence, solving quadratic equations requires systematic arithmetic and application of square root operations.
Isolating Variables
Isolating the variable is crucial for solving any equation, including quadratic ones. In simple terms, isolating the variable means to get the variable by itself on one side of the equation.
In the exercise provided, \(\frac{4}{3} x^{2}+2=110\), we began by isolating the term \(\frac{4}{3}x^2\). Here's how we did it:
  • Step 1: Subtract 2 from both sides. This gives us \(\frac{4}{3}x^2 = 108\).
  • Step 2: Multiply by the reciprocal of \(\frac{4}{3}\), which is \(\frac{3}{4}\). This simplification turns \(\frac{4}{3}x^2\) into 1\(x^2\).
These operations help us isolate \(x^2\). Now that we have \(x^2 = 81\), we move forward with solving for \(x\). Successfully isolating the variable sets you up for the final steps to solve the quadratic equation.
Square Roots
Taking the square root of both sides is the final step to determine the value of the variable in a simplified quadratic equation. The square root operation helps you find the two possible solutions for the variable when \(x^2\) is isolated.
In our exercise, after isolating \(x^2\), we have \(x^2 = 81\). To solve for \(x\), we take the square root of both sides:
  • \( \sqrt{x^2}= \sqrt{81}\). This operation directly provides us with \(x = 9\) or \(x = -9\).
Square roots are essential because each positive value for \(x\) has a corresponding negative value that satisfies the equation equally.
This is why you will notice the solutions in the form of \(\pm x\). Thus, our quadratic equation \(\frac{4}{3} x^{2}+2=110\) is solved with \(x\) being \(9\) and \(-9\).

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