Finite Math Examples

$p = q^{2} + 8 q + 16$ , $p = - 3 q^{2} + 6 q + 436$

Step 1

Eliminate the equal sides of each equation and combine.

$q^{2} + 8 q + 16 = - 3 q^{2} + 6 q + 436$

Step 2

Solve $q^{2} + 8 q + 16 = - 3 q^{2} + 6 q + 436$ for $q$ .

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Step 2.1

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

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Step 2.1.1

Add $3 q^{2}$ to both sides of the equation.

$q^{2} + 8 q + 16 + 3 q^{2} = 6 q + 436$

Step 2.1.2

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

$q^{2} + 8 q + 16 + 3 q^{2} - 6 q = 436$

Step 2.1.3

Add $q^{2}$ and $3 q^{2}$ .

$4 q^{2} + 8 q + 16 - 6 q = 436$

Step 2.1.4

Subtract $6 q$ from $8 q$ .

$4 q^{2} + 2 q + 16 = 436$

Step 2.2

Subtract $436$ from both sides of the equation.

$4 q^{2} + 2 q + 16 - 436 = 0$

Step 2.3

Subtract $436$ from $16$ .

$4 q^{2} + 2 q - 420 = 0$

Step 2.4

Factor the left side of the equation.

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Step 2.4.1

Factor $2$ out of $4 q^{2} + 2 q - 420$ .

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Step 2.4.1.1

Factor $2$ out of $4 q^{2}$ .

$2 (2 q^{2}) + 2 q - 420 = 0$

Step 2.4.1.2

Factor $2$ out of $2 q$ .

$2 (2 q^{2}) + 2 (q) - 420 = 0$

Step 2.4.1.3

Factor $2$ out of $- 420$ .

$2 (2 q^{2}) + 2 q + 2 \cdot - 210 = 0$

Step 2.4.1.4

Factor $2$ out of $2 (2 q^{2}) + 2 q$ .

$2 (2 q^{2} + q) + 2 \cdot - 210 = 0$

Step 2.4.1.5

Factor $2$ out of $2 (2 q^{2} + q) + 2 \cdot - 210$ .

$2 (2 q^{2} + q - 210) = 0$

Step 2.4.2

Factor.

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Step 2.4.2.1

Factor by grouping.

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Step 2.4.2.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 210 = - 420$ and whose sum is $b = 1$ .

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Step 2.4.2.1.1.1

Multiply by $1$ .

$2 (2 q^{2} + 1 q - 210) = 0$

Step 2.4.2.1.1.2

Rewrite $1$ as $- 20$ plus $21$

$2 (2 q^{2} + (- 20 + 21) q - 210) = 0$

Step 2.4.2.1.1.3

Apply the distributive property.

$2 (2 q^{2} - 20 q + 21 q - 210) = 0$

Step 2.4.2.1.2

Factor out the greatest common factor from each group.

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Step 2.4.2.1.2.1

Group the first two terms and the last two terms.

$2 ((2 q^{2} - 20 q) + 21 q - 210) = 0$

Step 2.4.2.1.2.2

Factor out the greatest common factor (GCF) from each group.

$2 (2 q (q - 10) + 21 (q - 10)) = 0$

Step 2.4.2.1.3

Factor the polynomial by factoring out the greatest common factor, $q - 10$ .

$2 ((q - 10) (2 q + 21)) = 0$

Step 2.4.2.2

Remove unnecessary parentheses.

$2 (q - 10) (2 q + 21) = 0$

Step 2.5

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$q - 10 = 0$

$2 q + 21 = 0 + p = - 3 q^{2} + 6 q + 436$

Step 2.6

Set $q - 10$ equal to $0$ and solve for $q$ .

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Step 2.6.1

Set $q - 10$ equal to $0$ .

$q - 10 = 0$

Step 2.6.2

Add $10$ to both sides of the equation.

$q = 10$

Step 2.7

Set $2 q + 21$ equal to $0$ and solve for $q$ .

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Step 2.7.1

Set $2 q + 21$ equal to $0$ .

$2 q + 21 = 0$

Step 2.7.2

Solve $2 q + 21 = 0$ for $q$ .

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Step 2.7.2.1

Subtract $21$ from both sides of the equation.

$2 q = - 21$

Step 2.7.2.2

Divide each term in $2 q = - 21$ by $2$ and simplify.

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Step 2.7.2.2.1

Divide each term in $2 q = - 21$ by $2$ .

$\frac{2 q}{2} = \frac{- 21}{2}$

Step 2.7.2.2.2

Simplify the left side.

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Step 2.7.2.2.2.1

Cancel the common factor of $2$ .

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Step 2.7.2.2.2.1.1

Cancel the common factor.

$\frac{2 q}{2} = \frac{- 21}{2}$

Step 2.7.2.2.2.1.2

Divide $q$ by $1$ .

$q = \frac{- 21}{2}$

Step 2.7.2.2.3

Simplify the right side.

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Step 2.7.2.2.3.1

Move the negative in front of the fraction.

$q = - \frac{21}{2}$

Step 2.8

The final solution is all the values that make $2 (q - 10) (2 q + 21) = 0$ true.

$q = 10, - \frac{21}{2}$

Step 3

Evaluate $p$ when $q = 10$ .

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Step 3.1

Substitute $10$ for $q$ .

$p = - 3 {(10)}^{2} + 6 (10) + 436$

Step 3.2

Substitute $10$ for $q$ in $p = - 3 {(10)}^{2} + 6 (10) + 436$ and solve for $p$ .

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Step 3.2.1

Remove parentheses.

$p = - 3 \cdot 10^{2} + 6 (10) + 436$

Step 3.2.2

Simplify $- 3 \cdot 10^{2} + 6 (10) + 436$ .

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Step 3.2.2.1

Raise $10$ to the power of $1$ .

$p = - 3 \cdot 10^{2} + 6 \cdot 10^{1} + 436$

Step 3.2.2.2

Move the decimal point in $6$ to the left by $1$ place and increase the power of $10^{1}$ by $1$ .

$p = - 3 \cdot 10^{2} + 0.6 \cdot 10^{2} + 436$

Step 3.2.2.3

Factor $10^{2}$ out of $- 3 \cdot 10^{2} + 0.6 \cdot 10^{2}$ .

$p = (- 3 + 0.6) \cdot 10^{2} + 436$

Step 3.2.2.4

Add $- 3$ and $0.6$ .

$p = - 2.4 \cdot 10^{2} + 436$

Step 3.2.2.5

Convert $436$ to scientific notation.

$p = - 2.4 \cdot 10^{2} + 4.36 \cdot 10^{2}$

Step 3.2.2.6

Factor $10^{2}$ out of $- 2.4 \cdot 10^{2} + 4.36 \cdot 10^{2}$ .

$p = (- 2.4 + 4.36) \cdot 10^{2}$

Step 3.2.2.7

Add $- 2.4$ and $4.36$ .

$p = 1.96 \cdot 10^{2}$

Step 4

Evaluate $p$ when $q = - \frac{21}{2}$ .

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Step 4.1

Substitute $- \frac{21}{2}$ for $q$ .

$p = - 3 {(- \frac{21}{2})}^{2} + 6 (- \frac{21}{2}) + 436$

Step 4.2

Simplify $- 3 {(- \frac{21}{2})}^{2} + 6 (- \frac{21}{2}) + 436$ .

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.2.1.1.1

Apply the product rule to $- \frac{21}{2}$ .

$p = - 3 ({(- 1)}^{2} {(\frac{21}{2})}^{2}) + 6 (- \frac{21}{2}) + 436$

Step 4.2.1.1.2

Apply the product rule to $\frac{21}{2}$ .

$p = - 3 ({(- 1)}^{2} \frac{21^{2}}{2^{2}}) + 6 (- \frac{21}{2}) + 436$

Step 4.2.1.2

Raise $- 1$ to the power of $2$ .

$p = - 3 (1 \frac{21^{2}}{2^{2}}) + 6 (- \frac{21}{2}) + 436$

Step 4.2.1.3

Multiply $\frac{21^{2}}{2^{2}}$ by $1$ .

$p = - 3 \frac{21^{2}}{2^{2}} + 6 (- \frac{21}{2}) + 436$

Step 4.2.1.4

Raise $21$ to the power of $2$ .

$p = - 3 \frac{441}{2^{2}} + 6 (- \frac{21}{2}) + 436$

Step 4.2.1.5

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

$p = - 3 (\frac{441}{4}) + 6 (- \frac{21}{2}) + 436$

Step 4.2.1.6

Multiply $- 3 (\frac{441}{4})$ .

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Step 4.2.1.6.1

Combine $- 3$ and $\frac{441}{4}$ .

$p = \frac{- 3 \cdot 441}{4} + 6 (- \frac{21}{2}) + 436$

Step 4.2.1.6.2

Multiply $- 3$ by $441$ .

$p = \frac{- 1323}{4} + 6 (- \frac{21}{2}) + 436$

Step 4.2.1.7

Move the negative in front of the fraction.

$p = - \frac{1323}{4} + 6 (- \frac{21}{2}) + 436$

Step 4.2.1.8

Cancel the common factor of $2$ .

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Step 4.2.1.8.1

Move the leading negative in $- \frac{21}{2}$ into the numerator.

$p = - \frac{1323}{4} + 6 (\frac{- 21}{2}) + 436$

Step 4.2.1.8.2

Factor $2$ out of $6$ .

$p = - \frac{1323}{4} + 2 (3) \frac{- 21}{2} + 436$

Step 4.2.1.8.3

Cancel the common factor.

$p = - \frac{1323}{4} + 2 \cdot 3 \frac{- 21}{2} + 436$

Step 4.2.1.8.4

Rewrite the expression.

$p = - \frac{1323}{4} + 3 \cdot - 21 + 436$

Step 4.2.1.9

Multiply $3$ by $- 21$ .

$p = - \frac{1323}{4} - 63 + 436$

Step 4.2.2

Find the common denominator.

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Step 4.2.2.1

Write $- 63$ as a fraction with denominator $1$ .

$p = - \frac{1323}{4} + \frac{- 63}{1} + 436$

Step 4.2.2.2

Multiply $\frac{- 63}{1}$ by $\frac{4}{4}$ .

$p = - \frac{1323}{4} + \frac{- 63}{1} \cdot \frac{4}{4} + 436$

Step 4.2.2.3

Multiply $\frac{- 63}{1}$ by $\frac{4}{4}$ .

$p = - \frac{1323}{4} + \frac{- 63 \cdot 4}{4} + 436$

Step 4.2.2.4

Write $436$ as a fraction with denominator $1$ .

$p = - \frac{1323}{4} + \frac{- 63 \cdot 4}{4} + \frac{436}{1}$

Step 4.2.2.5

Multiply $\frac{436}{1}$ by $\frac{4}{4}$ .

$p = - \frac{1323}{4} + \frac{- 63 \cdot 4}{4} + \frac{436}{1} \cdot \frac{4}{4}$

Step 4.2.2.6

Multiply $\frac{436}{1}$ by $\frac{4}{4}$ .

$p = - \frac{1323}{4} + \frac{- 63 \cdot 4}{4} + \frac{436 \cdot 4}{4}$

Step 4.2.3

Combine the numerators over the common denominator.

$p = \frac{- 1323 - 63 \cdot 4 + 436 \cdot 4}{4}$

Step 4.2.4

Simplify each term.

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Step 4.2.4.1

Multiply $- 63$ by $4$ .

$p = \frac{- 1323 - 252 + 436 \cdot 4}{4}$

Step 4.2.4.2

Multiply $436$ by $4$ .

$p = \frac{- 1323 - 252 + 1744}{4}$

Step 4.2.5

Simplify by adding and subtracting.

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Step 4.2.5.1

Subtract $252$ from $- 1323$ .

$p = \frac{- 1575 + 1744}{4}$

Step 4.2.5.2

Add $- 1575$ and $1744$ .

$p = \frac{169}{4}$

Step 5

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

$p = 1.96 \cdot 10^{2}, q = 10 p = \frac{169}{4}, q = - \frac{21}{2}$

Step 6

List all of the solutions.

$p = 1.96 \cdot 10^{2}, q = 10$

$p = \frac{169}{4}, q = - \frac{21}{2}$