Finite Math Examples

${(\sqrt{m + 18})}^{2} + 2 = {(m - 2)}^{2}$

Step 1

Simplify ${\sqrt{m + 18}}^{2} + 2$ .

Tap for more steps...

Step 1.1

Rewrite ${\sqrt{m + 18}}^{2}$ as $m + 18$ .

Tap for more steps...

Step 1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{m + 18}$ as ${(m + 18)}^{\frac{1}{2}}$ .

${({(m + 18)}^{\frac{1}{2}})}^{2} + 2 = {(m - 2)}^{2}$

Step 1.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(m + 18)}^{\frac{1}{2} \cdot 2} + 2 = {(m - 2)}^{2}$

Step 1.1.3

Combine $\frac{1}{2}$ and $2$ .

${(m + 18)}^{\frac{2}{2}} + 2 = {(m - 2)}^{2}$

Step 1.1.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.1.4.1

Cancel the common factor.

${(m + 18)}^{\frac{2}{2}} + 2 = {(m - 2)}^{2}$

Step 1.1.4.2

Rewrite the expression.

${(m + 18)}^{1} + 2 = {(m - 2)}^{2}$

Step 1.1.5

Simplify.

$m + 18 + 2 = {(m - 2)}^{2}$

Step 1.2

Add $18$ and $2$ .

$m + 20 = {(m - 2)}^{2}$

Step 2

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

Tap for more steps...

Step 2.1

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

$m + 20 = (m - 2) (m - 2)$

Step 2.2

Expand $(m - 2) (m - 2)$ using the FOIL Method.

Tap for more steps...

Step 2.2.1

Apply the distributive property.

$m + 20 = m (m - 2) - 2 (m - 2)$

Step 2.2.2

Apply the distributive property.

$m + 20 = m \cdot m + m \cdot - 2 - 2 (m - 2)$

Step 2.2.3

Apply the distributive property.

$m + 20 = m \cdot m + m \cdot - 2 - 2 m - 2 \cdot - 2$

Step 2.3

Simplify and combine like terms.

Tap for more steps...

Step 2.3.1

Simplify each term.

Tap for more steps...

Step 2.3.1.1

Multiply $m$ by $m$ .

$m + 20 = m^{2} + m \cdot - 2 - 2 m - 2 \cdot - 2$

Step 2.3.1.2

Move $- 2$ to the left of $m$ .

$m + 20 = m^{2} - 2 \cdot m - 2 m - 2 \cdot - 2$

Step 2.3.1.3

Multiply $- 2$ by $- 2$ .

$m + 20 = m^{2} - 2 m - 2 m + 4$

Step 2.3.2

Subtract $2 m$ from $- 2 m$ .

$m + 20 = m^{2} - 4 m + 4$

Step 3

Since $m$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$m^{2} - 4 m + 4 = m + 20$

Step 4

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

Tap for more steps...

Step 4.1

Subtract $m$ from both sides of the equation.

$m^{2} - 4 m + 4 - m = 20$

Step 4.2

Subtract $m$ from $- 4 m$ .

$m^{2} - 5 m + 4 = 20$

Step 5

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

Tap for more steps...

Step 5.1

Subtract $20$ from both sides of the equation.

$m^{2} - 5 m + 4 - 20 = 0$

Step 5.2

Subtract $20$ from $4$ .

$m^{2} - 5 m - 16 = 0$

Step 6

Use the quadratic formula to find the solutions.

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

Step 7

Substitute the values $a = 1$ , $b = - 5$ , and $c = - 16$ into the quadratic formula and solve for $m$ .

$\frac{5 \pm \sqrt{{(- 5)}^{2} - 4 \cdot (1 \cdot - 16)}}{2 \cdot 1}$

Step 8

Simplify.

Tap for more steps...

Step 8.1

Simplify the numerator.

Tap for more steps...

Step 8.1.1

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

$m = \frac{5 \pm \sqrt{25 - 4 \cdot 1 \cdot - 16}}{2 \cdot 1}$

Step 8.1.2

Multiply $- 4 \cdot 1 \cdot - 16$ .

Tap for more steps...

Step 8.1.2.1

Multiply $- 4$ by $1$ .

$m = \frac{5 \pm \sqrt{25 - 4 \cdot - 16}}{2 \cdot 1}$

Step 8.1.2.2

Multiply $- 4$ by $- 16$ .

$m = \frac{5 \pm \sqrt{25 + 64}}{2 \cdot 1}$

Step 8.1.3

Add $25$ and $64$ .

$m = \frac{5 \pm \sqrt{89}}{2 \cdot 1}$

Step 8.2

Multiply $2$ by $1$ .

$m = \frac{5 \pm \sqrt{89}}{2}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$m = \frac{5 \pm \sqrt{89}}{2}$

Decimal Form:

$m = 7.21699056 \dots, - 2.21699056 \dots$

Step 10