Finite Math Examples

${(5 m - 6)}^{2} = 7$

Step 1

Subtract $7$ from both sides of the equation.

${(5 m - 6)}^{2} - 7 = 0$

Step 2

Simplify ${(5 m - 6)}^{2} - 7$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite ${(5 m - 6)}^{2}$ as $(5 m - 6) (5 m - 6)$ .

$(5 m - 6) (5 m - 6) - 7 = 0$

Step 2.1.2

Expand $(5 m - 6) (5 m - 6)$ using the FOIL Method.

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Step 2.1.2.1

Apply the distributive property.

$5 m (5 m - 6) - 6 (5 m - 6) - 7 = 0$

Step 2.1.2.2

Apply the distributive property.

$5 m (5 m) + 5 m \cdot - 6 - 6 (5 m - 6) - 7 = 0$

Step 2.1.2.3

Apply the distributive property.

$5 m (5 m) + 5 m \cdot - 6 - 6 (5 m) - 6 \cdot - 6 - 7 = 0$

Step 2.1.3

Simplify and combine like terms.

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Step 2.1.3.1

Simplify each term.

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Step 2.1.3.1.1

Rewrite using the commutative property of multiplication.

$5 \cdot 5 m \cdot m + 5 m \cdot - 6 - 6 (5 m) - 6 \cdot - 6 - 7 = 0$

Step 2.1.3.1.2

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

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Step 2.1.3.1.2.1

Move $m$ .

$5 \cdot 5 (m \cdot m) + 5 m \cdot - 6 - 6 (5 m) - 6 \cdot - 6 - 7 = 0$

Step 2.1.3.1.2.2

Multiply $m$ by $m$ .

$5 \cdot 5 m^{2} + 5 m \cdot - 6 - 6 (5 m) - 6 \cdot - 6 - 7 = 0$

Step 2.1.3.1.3

Multiply $5$ by $5$ .

$25 m^{2} + 5 m \cdot - 6 - 6 (5 m) - 6 \cdot - 6 - 7 = 0$

Step 2.1.3.1.4

Multiply $- 6$ by $5$ .

$25 m^{2} - 30 m - 6 (5 m) - 6 \cdot - 6 - 7 = 0$

Step 2.1.3.1.5

Multiply $5$ by $- 6$ .

$25 m^{2} - 30 m - 30 m - 6 \cdot - 6 - 7 = 0$

Step 2.1.3.1.6

Multiply $- 6$ by $- 6$ .

$25 m^{2} - 30 m - 30 m + 36 - 7 = 0$

Step 2.1.3.2

Subtract $30 m$ from $- 30 m$ .

$25 m^{2} - 60 m + 36 - 7 = 0$

Step 2.2

Subtract $7$ from $36$ .

$25 m^{2} - 60 m + 29 = 0$

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = 25$ , $b = - 60$ , and $c = 29$ into the quadratic formula and solve for $m$ .

$\frac{60 \pm \sqrt{{(- 60)}^{2} - 4 \cdot (25 \cdot 29)}}{2 \cdot 25}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $- 60$ to the power of $2$ .

$m = \frac{60 \pm \sqrt{3600 - 4 \cdot 25 \cdot 29}}{2 \cdot 25}$

Step 5.1.2

Multiply $- 4 \cdot 25 \cdot 29$ .

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Step 5.1.2.1

Multiply $- 4$ by $25$ .

$m = \frac{60 \pm \sqrt{3600 - 100 \cdot 29}}{2 \cdot 25}$

Step 5.1.2.2

Multiply $- 100$ by $29$ .

$m = \frac{60 \pm \sqrt{3600 - 2900}}{2 \cdot 25}$

Step 5.1.3

Subtract $2900$ from $3600$ .

$m = \frac{60 \pm \sqrt{700}}{2 \cdot 25}$

Step 5.1.4

Rewrite $700$ as $10^{2} \cdot 7$ .

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Step 5.1.4.1

Factor $100$ out of $700$ .

$m = \frac{60 \pm \sqrt{100 (7)}}{2 \cdot 25}$

Step 5.1.4.2

Rewrite $100$ as $10^{2}$ .

$m = \frac{60 \pm \sqrt{10^{2} \cdot 7}}{2 \cdot 25}$

Step 5.1.5

Pull terms out from under the radical.

$m = \frac{60 \pm 10 \sqrt{7}}{2 \cdot 25}$

Step 5.2

Multiply $2$ by $25$ .

$m = \frac{60 \pm 10 \sqrt{7}}{50}$

Step 5.3

Simplify $\frac{60 \pm 10 \sqrt{7}}{50}$ .

$m = \frac{6 \pm \sqrt{7}}{5}$

Step 6

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $- 60$ to the power of $2$ .

$m = \frac{60 \pm \sqrt{3600 - 4 \cdot 25 \cdot 29}}{2 \cdot 25}$

Step 6.1.2

Multiply $- 4 \cdot 25 \cdot 29$ .

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Step 6.1.2.1

Multiply $- 4$ by $25$ .

$m = \frac{60 \pm \sqrt{3600 - 100 \cdot 29}}{2 \cdot 25}$

Step 6.1.2.2

Multiply $- 100$ by $29$ .

$m = \frac{60 \pm \sqrt{3600 - 2900}}{2 \cdot 25}$

Step 6.1.3

Subtract $2900$ from $3600$ .

$m = \frac{60 \pm \sqrt{700}}{2 \cdot 25}$

Step 6.1.4

Rewrite $700$ as $10^{2} \cdot 7$ .

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Step 6.1.4.1

Factor $100$ out of $700$ .

$m = \frac{60 \pm \sqrt{100 (7)}}{2 \cdot 25}$

Step 6.1.4.2

Rewrite $100$ as $10^{2}$ .

$m = \frac{60 \pm \sqrt{10^{2} \cdot 7}}{2 \cdot 25}$

Step 6.1.5

Pull terms out from under the radical.

$m = \frac{60 \pm 10 \sqrt{7}}{2 \cdot 25}$

Step 6.2

Multiply $2$ by $25$ .

$m = \frac{60 \pm 10 \sqrt{7}}{50}$

Step 6.3

Simplify $\frac{60 \pm 10 \sqrt{7}}{50}$ .

$m = \frac{6 \pm \sqrt{7}}{5}$

Step 6.4

Change the $\pm$ to $+$ .

$m = \frac{6 + \sqrt{7}}{5}$

Step 7

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $- 60$ to the power of $2$ .

$m = \frac{60 \pm \sqrt{3600 - 4 \cdot 25 \cdot 29}}{2 \cdot 25}$

Step 7.1.2

Multiply $- 4 \cdot 25 \cdot 29$ .

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Step 7.1.2.1

Multiply $- 4$ by $25$ .

$m = \frac{60 \pm \sqrt{3600 - 100 \cdot 29}}{2 \cdot 25}$

Step 7.1.2.2

Multiply $- 100$ by $29$ .

$m = \frac{60 \pm \sqrt{3600 - 2900}}{2 \cdot 25}$

Step 7.1.3

Subtract $2900$ from $3600$ .

$m = \frac{60 \pm \sqrt{700}}{2 \cdot 25}$

Step 7.1.4

Rewrite $700$ as $10^{2} \cdot 7$ .

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Step 7.1.4.1

Factor $100$ out of $700$ .

$m = \frac{60 \pm \sqrt{100 (7)}}{2 \cdot 25}$

Step 7.1.4.2

Rewrite $100$ as $10^{2}$ .

$m = \frac{60 \pm \sqrt{10^{2} \cdot 7}}{2 \cdot 25}$

Step 7.1.5

Pull terms out from under the radical.

$m = \frac{60 \pm 10 \sqrt{7}}{2 \cdot 25}$

Step 7.2

Multiply $2$ by $25$ .

$m = \frac{60 \pm 10 \sqrt{7}}{50}$

Step 7.3

Simplify $\frac{60 \pm 10 \sqrt{7}}{50}$ .

$m = \frac{6 \pm \sqrt{7}}{5}$

Step 7.4

Change the $\pm$ to $-$ .

$m = \frac{6 - \sqrt{7}}{5}$

Step 8

The final answer is the combination of both solutions.

$m = \frac{6 + \sqrt{7}}{5}, \frac{6 - \sqrt{7}}{5}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$m = \frac{6 + \sqrt{7}}{5}, \frac{6 - \sqrt{7}}{5}$

Decimal Form:

$m = 1.72915026 \dots, 0.67084973 \dots$