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Chapter 9: Problem 20

\(-\frac{1}{2} x^{2}=-18\)

Short Answer

Expert verified
x = 6 or x = -6

Step by step solution

01

Isolate the variable term

To start, we need to isolate the term with the variable. The equation is \( -\frac{1}{2} x^{2}=-18 \). Multiply both sides of the equation by -2 to cancel out the fraction: \( x^2 = -18 \times -2 \). This simplifies to \( x^2 = 36 \).
02

Solve for x

Now that we have \( x^2 = 36 \), we take the square root of both sides to solve for x: \( x = \pm \sqrt{36} \). Thus, \( x = \pm 6 \).

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Key Concepts

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

Isolating Variable Terms
In solving quadratic equations, our goal is to identify the variable term alone on one side of the equation.
For example, consider the problem \(-\frac{1}{2} x^{2} = -18\).
To isolate the variable term, we need to eliminate the coefficient.
In this problem, the coefficient is \-\frac{1}{2}\.
To cancel out \-\frac{1}{2}\, multiply both sides of the equation by its reciprocal, which is -2:
\( x^2 = -18 \times -2 \).
Multiplying, we get:
\( x^2 = 36 \).
Now, the variable term \( x^2 \) is isolated on one side of the equation.
Multiplying Both Sides of the Equation
When isolating variable terms, we often need to multiply both sides of the equation.
This keeps the equation balanced.
For instance, in the equation \(-\frac{1}{2} x^{2} = -18\), the fraction \-\frac{1}{2}\ makes it more complex.
To simplify, multiply both sides by -2.
This step eliminates the fraction and simplifies our equation.
Here鈥檚 how it looks:
\( -\frac{1}{2} x^{2} \times -2 = -18 \times -2 \).
After this multiplication, we get: \( x^2 = 36 \).
This step is crucial because it transforms the equation into a simpler form.
Taking Square Roots
The final main concept is taking the square root to solve for the variable.
Once you have isolated the variable term, you can solve for the variable by taking the square root of both sides.
In our example, we have \( x^2 = 36 \).
To solve for x, we take the square root of 36.
The square root of 36 is 6, but remember, there are always two possible solutions for \( x^2 = 36 \): positive and negative.
Thus, we write: \( x = \pm \sqrt{36} \), which simplifies to \( x = \pm 6 \).
So, the final solutions for x are: \( x = 6 \) or \( x = -6 \).
Taking the square root is a powerful step in solving quadratic equations because it directly gives us the values for the variable.

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Most popular questions from this chapter

\(n^{2}+5 n+6=0\)

The diameter at basal height of a tree is the diameter about \(3 \mathrm{ft}\) above the ground. The approximate shape of the trunk of a tree above basal height is a cone. The formula for the volume of a cone is \(V=\frac{\pi r^{2} h}{3}\), where \(r\) is the radius and \(h\) is the height. Measured from its basal height, a tree is \(50 \mathrm{ft}\) tall. The diameter of its trunk is \(2.5 \mathrm{ft}\). Find the approximate volume of lumber in cubic feet in this trunk. ( \(\pi \approx 3.14\).) Round to the nearest whole number. (Source: G. John Smith; www.math.bcit.ca, 1997)

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