Finite Math Examples

${(2 m + 10)}^{2} + {(3 m)}^{2} = {(5 m)}^{2}$

Step 1

Simplify each term.

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Step 1.1

Apply the product rule to $3 m$ .

${(2 m + 10)}^{2} + 3^{2} m^{2} = {(5 m)}^{2}$

Step 1.2

Raise $3$ to the power of $2$ .

${(2 m + 10)}^{2} + 9 m^{2} = {(5 m)}^{2}$

Step 2

Simplify ${(5 m)}^{2}$ .

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Step 2.1

Apply the product rule to $5 m$ .

${(2 m + 10)}^{2} + 9 m^{2} = 5^{2} m^{2}$

Step 2.2

Raise $5$ to the power of $2$ .

${(2 m + 10)}^{2} + 9 m^{2} = 25 m^{2}$

Step 3

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

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Step 3.1

Subtract $25 m^{2}$ from both sides of the equation.

${(2 m + 10)}^{2} + 9 m^{2} - 25 m^{2} = 0$

Step 3.2

Simplify each term.

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Step 3.2.1

Rewrite ${(2 m + 10)}^{2}$ as $(2 m + 10) (2 m + 10)$ .

$(2 m + 10) (2 m + 10) + 9 m^{2} - 25 m^{2} = 0$

Step 3.2.2

Expand $(2 m + 10) (2 m + 10)$ using the FOIL Method.

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Step 3.2.2.1

Apply the distributive property.

$2 m (2 m + 10) + 10 (2 m + 10) + 9 m^{2} - 25 m^{2} = 0$

Step 3.2.2.2

Apply the distributive property.

$2 m (2 m) + 2 m \cdot 10 + 10 (2 m + 10) + 9 m^{2} - 25 m^{2} = 0$

Step 3.2.2.3

Apply the distributive property.

$2 m (2 m) + 2 m \cdot 10 + 10 (2 m) + 10 \cdot 10 + 9 m^{2} - 25 m^{2} = 0$

Step 3.2.3

Simplify and combine like terms.

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Step 3.2.3.1

Simplify each term.

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Step 3.2.3.1.1

Rewrite using the commutative property of multiplication.

$2 \cdot 2 m \cdot m + 2 m \cdot 10 + 10 (2 m) + 10 \cdot 10 + 9 m^{2} - 25 m^{2} = 0$

Step 3.2.3.1.2

Multiply $m$ by $m$ by adding the exponents.

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Step 3.2.3.1.2.1

Move $m$ .

$2 \cdot 2 (m \cdot m) + 2 m \cdot 10 + 10 (2 m) + 10 \cdot 10 + 9 m^{2} - 25 m^{2} = 0$

Step 3.2.3.1.2.2

Multiply $m$ by $m$ .

$2 \cdot 2 m^{2} + 2 m \cdot 10 + 10 (2 m) + 10 \cdot 10 + 9 m^{2} - 25 m^{2} = 0$

Step 3.2.3.1.3

Multiply $2$ by $2$ .

$4 m^{2} + 2 m \cdot 10 + 10 (2 m) + 10 \cdot 10 + 9 m^{2} - 25 m^{2} = 0$

Step 3.2.3.1.4

Multiply $10$ by $2$ .

$4 m^{2} + 20 m + 10 (2 m) + 10 \cdot 10 + 9 m^{2} - 25 m^{2} = 0$

Step 3.2.3.1.5

Multiply $2$ by $10$ .

$4 m^{2} + 20 m + 20 m + 10 \cdot 10 + 9 m^{2} - 25 m^{2} = 0$

Step 3.2.3.1.6

Multiply $10$ by $10$ .

$4 m^{2} + 20 m + 20 m + 100 + 9 m^{2} - 25 m^{2} = 0$

Step 3.2.3.2

Add $20 m$ and $20 m$ .

$4 m^{2} + 40 m + 100 + 9 m^{2} - 25 m^{2} = 0$

Step 3.3

Add $4 m^{2}$ and $9 m^{2}$ .

$13 m^{2} + 40 m + 100 - 25 m^{2} = 0$

Step 3.4

Subtract $25 m^{2}$ from $13 m^{2}$ .

$- 12 m^{2} + 40 m + 100 = 0$

Step 4

Factor the left side of the equation.

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Step 4.1

Factor $- 4$ out of $- 12 m^{2} + 40 m + 100$ .

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Step 4.1.1

Factor $- 4$ out of $- 12 m^{2}$ .

$- 4 (3 m^{2}) + 40 m + 100 = 0$

Step 4.1.2

Factor $- 4$ out of $40 m$ .

$- 4 (3 m^{2}) - 4 (- 10 m) + 100 = 0$

Step 4.1.3

Factor $- 4$ out of $100$ .

$- 4 (3 m^{2}) - 4 (- 10 m) - 4 \cdot - 25 = 0$

Step 4.1.4

Factor $- 4$ out of $- 4 (3 m^{2}) - 4 (- 10 m)$ .

$- 4 (3 m^{2} - 10 m) - 4 \cdot - 25 = 0$

Step 4.1.5

Factor $- 4$ out of $- 4 (3 m^{2} - 10 m) - 4 (- 25)$ .

$- 4 (3 m^{2} - 10 m - 25) = 0$

Step 4.2

Factor.

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Step 4.2.1

Factor by grouping.

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Step 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 $a \cdot c = 3 \cdot - 25 = - 75$ and whose sum is $b = - 10$ .

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Step 4.2.1.1.1

Factor $- 10$ out of $- 10 m$ .

$- 4 (3 m^{2} - 10 m - 25) = 0$

Step 4.2.1.1.2

Rewrite $- 10$ as $5$ plus $- 15$

$- 4 (3 m^{2} + (5 - 15) m - 25) = 0$

Step 4.2.1.1.3

Apply the distributive property.

$- 4 (3 m^{2} + 5 m - 15 m - 25) = 0$

Step 4.2.1.2

Factor out the greatest common factor from each group.

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Step 4.2.1.2.1

Group the first two terms and the last two terms.

$- 4 ((3 m^{2} + 5 m) - 15 m - 25) = 0$

Step 4.2.1.2.2

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

$- 4 (m (3 m + 5) - 5 (3 m + 5)) = 0$

Step 4.2.1.3

Factor the polynomial by factoring out the greatest common factor, $3 m + 5$ .

$- 4 ((3 m + 5) (m - 5)) = 0$

Step 4.2.2

Remove unnecessary parentheses.

$- 4 (3 m + 5) (m - 5) = 0$

Step 5

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

$3 m + 5 = 0$

$m - 5 = 0$

Step 6

Set $3 m + 5$ equal to $0$ and solve for $m$ .

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Step 6.1

Set $3 m + 5$ equal to $0$ .

$3 m + 5 = 0$

Step 6.2

Solve $3 m + 5 = 0$ for $m$ .

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Step 6.2.1

Subtract $5$ from both sides of the equation.

$3 m = - 5$

Step 6.2.2

Divide each term in $3 m = - 5$ by $3$ and simplify.

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Step 6.2.2.1

Divide each term in $3 m = - 5$ by $3$ .

$\frac{3 m}{3} = \frac{- 5}{3}$

Step 6.2.2.2

Simplify the left side.

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Step 6.2.2.2.1

Cancel the common factor of $3$ .

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Step 6.2.2.2.1.1

Cancel the common factor.

$\frac{3 m}{3} = \frac{- 5}{3}$

Step 6.2.2.2.1.2

Divide $m$ by $1$ .

$m = \frac{- 5}{3}$

Step 6.2.2.3

Simplify the right side.

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Step 6.2.2.3.1

Move the negative in front of the fraction.

$m = - \frac{5}{3}$

Step 7

Set $m - 5$ equal to $0$ and solve for $m$ .

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Step 7.1

Set $m - 5$ equal to $0$ .

$m - 5 = 0$

Step 7.2

Add $5$ to both sides of the equation.

$m = 5$

Step 8

The final solution is all the values that make $- 4 (3 m + 5) (m - 5) = 0$ true.

$m = - \frac{5}{3}, 5$

Step 9