Calculus Examples

$144 - x^{2} = 48 + \frac{1}{2} x^{2}$

Step 1

Combine $\frac{1}{2}$ and $x^{2}$ .

$144 - x^{2} = 48 + \frac{x^{2}}{2}$

Step 2

Move all terms containing $x$ to the left side of the equation.

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Step 2.1

Subtract $\frac{x^{2}}{2}$ from both sides of the equation.

$144 - x^{2} - \frac{x^{2}}{2} = 48$

Step 2.2

To write $- x^{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$144 - x^{2} \cdot \frac{2}{2} - \frac{x^{2}}{2} = 48$

Step 2.3

Combine $- x^{2}$ and $\frac{2}{2}$ .

$144 + \frac{- x^{2} \cdot 2}{2} - \frac{x^{2}}{2} = 48$

Step 2.4

Combine the numerators over the common denominator.

$144 + \frac{- x^{2} \cdot 2 - x^{2}}{2} = 48$

Step 2.5

Simplify each term.

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Step 2.5.1

Simplify the numerator.

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Step 2.5.1.1

Factor $x^{2}$ out of $- x^{2} \cdot 2 - x^{2}$ .

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Step 2.5.1.1.1

Factor $x^{2}$ out of $- x^{2} \cdot 2$ .

$144 + \frac{x^{2} (- 1 \cdot 2) - x^{2}}{2} = 48$

Step 2.5.1.1.2

Factor $x^{2}$ out of $- x^{2}$ .

$144 + \frac{x^{2} (- 1 \cdot 2) + x^{2} \cdot - 1}{2} = 48$

Step 2.5.1.1.3

Factor $x^{2}$ out of $x^{2} (- 1 \cdot 2) + x^{2} \cdot - 1$ .

$144 + \frac{x^{2} (- 1 \cdot 2 - 1)}{2} = 48$

Step 2.5.1.2

Multiply $- 1$ by $2$ .

$144 + \frac{x^{2} (- 2 - 1)}{2} = 48$

Step 2.5.1.3

Subtract $1$ from $- 2$ .

$144 + \frac{x^{2} \cdot - 3}{2} = 48$

Step 2.5.2

Move $- 3$ to the left of $x^{2}$ .

$144 + \frac{- 3 \cdot x^{2}}{2} = 48$

Step 2.5.3

Move the negative in front of the fraction.

$144 - \frac{3 x^{2}}{2} = 48$

Step 3

Move all terms not containing $x$ to the right side of the equation.

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Step 3.1

Subtract $144$ from both sides of the equation.

$- \frac{3 x^{2}}{2} = 48 - 144$

Step 3.2

Subtract $144$ from $48$ .

$- \frac{3 x^{2}}{2} = - 96$

Step 4

Multiply both sides of the equation by $- \frac{2}{3}$ .

$- \frac{2}{3} (- \frac{3 x^{2}}{2}) = - \frac{2}{3} \cdot - 96$

Step 5

Simplify both sides of the equation.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Simplify $- \frac{2}{3} (- \frac{3 x^{2}}{2})$ .

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Step 5.1.1.1

Cancel the common factor of $2$ .

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Step 5.1.1.1.1

Move the leading negative in $- \frac{2}{3}$ into the numerator.

$\frac{- 2}{3} (- \frac{3 x^{2}}{2}) = - \frac{2}{3} \cdot - 96$

Step 5.1.1.1.2

Move the leading negative in $- \frac{3 x^{2}}{2}$ into the numerator.

$\frac{- 2}{3} \cdot \frac{- 3 x^{2}}{2} = - \frac{2}{3} \cdot - 96$

Step 5.1.1.1.3

Factor $2$ out of $- 2$ .

$\frac{2 (- 1)}{3} \cdot \frac{- 3 x^{2}}{2} = - \frac{2}{3} \cdot - 96$

Step 5.1.1.1.4

Cancel the common factor.

$\frac{2 \cdot - 1}{3} \cdot \frac{- 3 x^{2}}{2} = - \frac{2}{3} \cdot - 96$

Step 5.1.1.1.5

Rewrite the expression.

$\frac{- 1}{3} (- 3 x^{2}) = - \frac{2}{3} \cdot - 96$

Step 5.1.1.2

Cancel the common factor of $3$ .

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Step 5.1.1.2.1

Factor $3$ out of $- 3 x^{2}$ .

$\frac{- 1}{3} (3 (- x^{2})) = - \frac{2}{3} \cdot - 96$

Step 5.1.1.2.2

Cancel the common factor.

$\frac{- 1}{3} (3 (- x^{2})) = - \frac{2}{3} \cdot - 96$

Step 5.1.1.2.3

Rewrite the expression.

$- - x^{2} = - \frac{2}{3} \cdot - 96$

Step 5.1.1.3

Multiply.

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Step 5.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 x^{2} = - \frac{2}{3} \cdot - 96$

Step 5.1.1.3.2

Multiply $x^{2}$ by $1$ .

$x^{2} = - \frac{2}{3} \cdot - 96$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $- \frac{2}{3} \cdot - 96$ .

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Step 5.2.1.1

Cancel the common factor of $3$ .

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Step 5.2.1.1.1

Move the leading negative in $- \frac{2}{3}$ into the numerator.

$x^{2} = \frac{- 2}{3} \cdot - 96$

Step 5.2.1.1.2

Factor $3$ out of $- 96$ .

$x^{2} = \frac{- 2}{3} \cdot (3 (- 32))$

Step 5.2.1.1.3

Cancel the common factor.

$x^{2} = \frac{- 2}{3} \cdot (3 \cdot - 32)$

Step 5.2.1.1.4

Rewrite the expression.

$x^{2} = - 2 \cdot - 32$

Step 5.2.1.2

Multiply $- 2$ by $- 32$ .

$x^{2} = 64$

Step 6

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{64}$

Step 7

Simplify $\pm \sqrt{64}$ .

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Step 7.1

Rewrite $64$ as $8^{2}$ .

$x = \pm \sqrt{8^{2}}$

Step 7.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 8$

Step 8

The complete solution is the result of both the positive and negative portions of the solution.

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Step 8.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 8$

Step 8.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 8$

Step 8.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 8, - 8$