Basic Math Examples

$(12 + p) (9 - p) = 28 + 6 p$

Step 1

Simplify $(12 + p) (9 - p)$ .

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Step 1.1

Rewrite.

$0 + 0 + (12 + p) (9 - p) = 28 + 6 p$

Step 1.2

Simplify by adding zeros.

$(12 + p) (9 - p) = 28 + 6 p$

Step 1.3

Expand $(12 + p) (9 - p)$ using the FOIL Method.

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Step 1.3.1

Apply the distributive property.

$12 (9 - p) + p (9 - p) = 28 + 6 p$

Step 1.3.2

Apply the distributive property.

$12 \cdot 9 + 12 (- p) + p (9 - p) = 28 + 6 p$

Step 1.3.3

Apply the distributive property.

$12 \cdot 9 + 12 (- p) + p \cdot 9 + p (- p) = 28 + 6 p$

Step 1.4

Simplify and combine like terms.

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Step 1.4.1

Simplify each term.

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Step 1.4.1.1

Multiply $12$ by $9$ .

$108 + 12 (- p) + p \cdot 9 + p (- p) = 28 + 6 p$

Step 1.4.1.2

Multiply $- 1$ by $12$ .

$108 - 12 p + p \cdot 9 + p (- p) = 28 + 6 p$

Step 1.4.1.3

Move $9$ to the left of $p$ .

$108 - 12 p + 9 \cdot p + p (- p) = 28 + 6 p$

Step 1.4.1.4

Rewrite using the commutative property of multiplication.

$108 - 12 p + 9 p - p \cdot p = 28 + 6 p$

Step 1.4.1.5

Multiply $p$ by $p$ by adding the exponents.

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Step 1.4.1.5.1

Move $p$ .

$108 - 12 p + 9 p - (p \cdot p) = 28 + 6 p$

Step 1.4.1.5.2

Multiply $p$ by $p$ .

$108 - 12 p + 9 p - p^{2} = 28 + 6 p$

Step 1.4.2

Add $- 12 p$ and $9 p$ .

$108 - 3 p - p^{2} = 28 + 6 p$

Step 2

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

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Step 2.1

Subtract $6 p$ from both sides of the equation.

$108 - 3 p - p^{2} - 6 p = 28$

Step 2.2

Subtract $6 p$ from $- 3 p$ .

$108 - p^{2} - 9 p = 28$

Step 3

Move all terms to the left side of the equation and simplify.

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Step 3.1

Subtract $28$ from both sides of the equation.

$108 - p^{2} - 9 p - 28 = 0$

Step 3.2

Subtract $28$ from $108$ .

$- p^{2} - 9 p + 80 = 0$

Step 4

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 5

Substitute the values $a = - 1$ , $b = - 9$ , and $c = 80$ into the quadratic formula and solve for $p$ .

$\frac{9 \pm \sqrt{{(- 9)}^{2} - 4 \cdot (- 1 \cdot 80)}}{2 \cdot - 1}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $- 9$ to the power of $2$ .

$p = \frac{9 \pm \sqrt{81 - 4 \cdot - 1 \cdot 80}}{2 \cdot - 1}$

Step 6.1.2

Multiply $- 4 \cdot - 1 \cdot 80$ .

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Step 6.1.2.1

Multiply $- 4$ by $- 1$ .

$p = \frac{9 \pm \sqrt{81 + 4 \cdot 80}}{2 \cdot - 1}$

Step 6.1.2.2

Multiply $4$ by $80$ .

$p = \frac{9 \pm \sqrt{81 + 320}}{2 \cdot - 1}$

Step 6.1.3

Add $81$ and $320$ .

$p = \frac{9 \pm \sqrt{401}}{2 \cdot - 1}$

Step 6.2

Multiply $2$ by $- 1$ .

$p = \frac{9 \pm \sqrt{401}}{- 2}$

Step 6.3

Move the negative in front of the fraction.

$p = - \frac{9 \pm \sqrt{401}}{2}$

Step 7

The final answer is the combination of both solutions.

$p = - \frac{9 + \sqrt{401}}{2}, - \frac{9 - \sqrt{401}}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$p = - \frac{9 + \sqrt{401}}{2}, - \frac{9 - \sqrt{401}}{2}$

Decimal Form:

$p = - 14.51249219 \dots, 5.51249219 \dots$