Precalculus Examples

$5 p^{2} + 20 p + 25 = 10$

Step 1

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

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Step 1.1

Subtract $25$ from both sides of the equation.

$5 p^{2} + 20 p = 10 - 25$

Step 1.2

Subtract $25$ from $10$ .

$5 p^{2} + 20 p = - 15$

Step 2

Divide each term in $5 p^{2} + 20 p = - 15$ by $5$ and simplify.

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Step 2.1

Divide each term in $5 p^{2} + 20 p = - 15$ by $5$ .

$\frac{5 p^{2}}{5} + \frac{20 p}{5} = \frac{- 15}{5}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Cancel the common factor of $5$ .

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Step 2.2.1.1.1

Cancel the common factor.

$\frac{5 p^{2}}{5} + \frac{20 p}{5} = \frac{- 15}{5}$

Step 2.2.1.1.2

Divide $p^{2}$ by $1$ .

$p^{2} + \frac{20 p}{5} = \frac{- 15}{5}$

Step 2.2.1.2

Cancel the common factor of $20$ and $5$ .

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Step 2.2.1.2.1

Factor $5$ out of $20 p$ .

$p^{2} + \frac{5 (4 p)}{5} = \frac{- 15}{5}$

Step 2.2.1.2.2

Cancel the common factors.

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Step 2.2.1.2.2.1

Factor $5$ out of $5$ .

$p^{2} + \frac{5 (4 p)}{5 (1)} = \frac{- 15}{5}$

Step 2.2.1.2.2.2

Cancel the common factor.

$p^{2} + \frac{5 (4 p)}{5 \cdot 1} = \frac{- 15}{5}$

Step 2.2.1.2.2.3

Rewrite the expression.

$p^{2} + \frac{4 p}{1} = \frac{- 15}{5}$

Step 2.2.1.2.2.4

Divide $4 p$ by $1$ .

$p^{2} + 4 p = \frac{- 15}{5}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $- 15$ by $5$ .

$p^{2} + 4 p = - 3$

Step 3

To create a trinomial square on the left side of the equation, find a value that is equal to the square of half of $b$ .

${(\frac{b}{2})}^{2} = {(2)}^{2}$

Step 4

Add the term to each side of the equation.

$p^{2} + 4 p + {(2)}^{2} = - 3 + {(2)}^{2}$

Step 5

Simplify the equation.

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Step 5.1

Simplify the left side.

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Step 5.1.1

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

$p^{2} + 4 p + 4 = - 3 + {(2)}^{2}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $- 3 + {(2)}^{2}$ .

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Step 5.2.1.1

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

$p^{2} + 4 p + 4 = - 3 + 4$

Step 5.2.1.2

Add $- 3$ and $4$ .

$p^{2} + 4 p + 4 = 1$

Step 6

Factor the perfect trinomial square into ${(p + 2)}^{2}$ .

${(p + 2)}^{2} = 1$

Step 7

Solve the equation for $p$ .

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Step 7.1

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

$p + 2 = \pm \sqrt{1}$

Step 7.2

Any root of $1$ is $1$ .

$p + 2 = \pm 1$

Step 7.3

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

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Step 7.3.1

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

$p + 2 = 1$

Step 7.3.2

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

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Step 7.3.2.1

Subtract $2$ from both sides of the equation.

$p = 1 - 2$

Step 7.3.2.2

Subtract $2$ from $1$ .

$p = - 1$

Step 7.3.3

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

$p + 2 = - 1$

Step 7.3.4

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

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Step 7.3.4.1

Subtract $2$ from both sides of the equation.

$p = - 1 - 2$

Step 7.3.4.2

Subtract $2$ from $- 1$ .

$p = - 3$

Step 7.3.5

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

$p = - 1, - 3$