Precalculus Examples

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

Step 1

Eliminate the equal sides of each equation and combine.

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

Step 2

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

Tap for more steps...

Step 2.1

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

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

Step 2.2

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

Tap for more steps...

Step 2.2.1

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

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

Step 2.2.2

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 2.2.3

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

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

Step 2.2.4

Combine the numerators over the common denominator.

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

Step 2.2.5

Simplify each term.

Tap for more steps...

Step 2.2.5.1

Simplify the numerator.

Tap for more steps...

Step 2.2.5.1.1

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

Tap for more steps...

Step 2.2.5.1.1.1

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

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

Step 2.2.5.1.1.2

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

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

Step 2.2.5.1.1.3

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 2.2.5.1.2

Multiply $- 1$ by $2$ .

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

Step 2.2.5.1.3

Subtract $1$ from $- 2$ .

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

Step 2.2.5.2

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

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

Step 2.2.5.3

Move the negative in front of the fraction.

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

Step 2.3

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

Tap for more steps...

Step 2.3.1

Subtract $144$ from both sides of the equation.

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

Step 2.3.2

Subtract $144$ from $48$ .

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

Step 2.4

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

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

Step 2.5

Simplify both sides of the equation.

Tap for more steps...

Step 2.5.1

Simplify the left side.

Tap for more steps...

Step 2.5.1.1

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

Tap for more steps...

Step 2.5.1.1.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.5.1.1.1.1

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 2.5.1.1.1.2

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 2.5.1.1.1.3

Factor $2$ out of $- 2$ .

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

Step 2.5.1.1.1.4

Cancel the common factor.

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

Step 2.5.1.1.1.5

Rewrite the expression.

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

Step 2.5.1.1.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 2.5.1.1.2.1

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

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

Step 2.5.1.1.2.2

Cancel the common factor.

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

Step 2.5.1.1.2.3

Rewrite the expression.

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

Step 2.5.1.1.3

Multiply.

Tap for more steps...

Step 2.5.1.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 2.5.1.1.3.2

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

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

Step 2.5.2

Simplify the right side.

Tap for more steps...

Step 2.5.2.1

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

Tap for more steps...

Step 2.5.2.1.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 2.5.2.1.1.1

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

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

Step 2.5.2.1.1.2

Factor $3$ out of $- 96$ .

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

Step 2.5.2.1.1.3

Cancel the common factor.

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

Step 2.5.2.1.1.4

Rewrite the expression.

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

Step 2.5.2.1.2

Multiply $- 2$ by $- 32$ .

$x^{2} = 64$

Step 2.6

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

$x = \pm \sqrt{64}$

Step 2.7

Simplify $\pm \sqrt{64}$ .

Tap for more steps...

Step 2.7.1

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

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

Step 2.7.2

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

$x = \pm 8$

Step 2.8

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

Tap for more steps...

Step 2.8.1

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

$x = 8$

Step 2.8.2

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

$x = - 8$

Step 2.8.3

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

$x = 8, - 8$

Step 3

Evaluate $p$ when $x = 8$ .

Tap for more steps...

Step 3.1

Substitute $8$ for $x$ .

$p = 48 + \frac{1}{2} \cdot {(8)}^{2}$

Step 3.2

Substitute $8$ for $x$ in $p = 48 + \frac{1}{2} \cdot {(8)}^{2}$ and solve for $p$ .

Tap for more steps...

Step 3.2.1

Remove parentheses.

$p = 48 + \frac{1}{2} \cdot 8^{2}$

Step 3.2.2

Simplify $48 + \frac{1}{2} \cdot 8^{2}$ .

Tap for more steps...

Step 3.2.2.1

Simplify each term.

Tap for more steps...

Step 3.2.2.1.1

Raise $8$ to the power of $2$ .

$p = 48 + \frac{1}{2} \cdot 64$

Step 3.2.2.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.2.2.1.2.1

Factor $2$ out of $64$ .

$p = 48 + \frac{1}{2} \cdot (2 (32))$

Step 3.2.2.1.2.2

Cancel the common factor.

$p = 48 + \frac{1}{2} \cdot (2 \cdot 32)$

Step 3.2.2.1.2.3

Rewrite the expression.

$p = 48 + 32$

Step 3.2.2.2

Add $48$ and $32$ .

$p = 80$

Step 4

The solution of the system of equations is all the values that make the system true.

$p = 80, x = 8 p = 80, x = - 8$

Step 5

List all of the solutions.

$p = 80, x = 8$

$p = 80, x = - 8$