Solve for m m+4 = square root of m+10

$m + 4 = \sqrt{m + 10}$

Step 1

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

$\sqrt{m + 10} = m + 4$

Step 2

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{m + 10}}^{2} = {(m + 4)}^{2}$

Step 3

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{m + 10}$ as ${(m + 10)}^{\frac{1}{2}}$ .

${({(m + 10)}^{\frac{1}{2}})}^{2} = {(m + 4)}^{2}$

Simplify the left side.

Simplify ${({(m + 10)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(m + 10)}^{\frac{1}{2}})}^{2}$ .

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

${(m + 10)}^{\frac{1}{2} \cdot 2} = {(m + 4)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(m + 10)}^{\frac{1}{2} \cdot 2} = {(m + 4)}^{2}$

Rewrite the expression.

${(m + 10)}^{1} = {(m + 4)}^{2}$

Simplify.

$m + 10 = {(m + 4)}^{2}$

Simplify the right side.

Simplify ${(m + 4)}^{2}$ .

Rewrite ${(m + 4)}^{2}$ as $(m + 4) (m + 4)$ .

$m + 10 = (m + 4) (m + 4)$

Expand $(m + 4) (m + 4)$ using the FOIL Method.

Apply the distributive property.

$m + 10 = m (m + 4) + 4 (m + 4)$

Apply the distributive property.

$m + 10 = m \cdot m + m \cdot 4 + 4 (m + 4)$

Apply the distributive property.

$m + 10 = m \cdot m + m \cdot 4 + 4 m + 4 \cdot 4$

Simplify and combine like terms.

Simplify each term.

Multiply $m$ by $m$ .

$m + 10 = m^{2} + m \cdot 4 + 4 m + 4 \cdot 4$

Move $4$ to the left of $m$ .

$m + 10 = m^{2} + 4 \cdot m + 4 m + 4 \cdot 4$

Multiply $4$ by $4$ .

$m + 10 = m^{2} + 4 m + 4 m + 16$

Add $4 m$ and $4 m$ .

$m + 10 = m^{2} + 8 m + 16$

Step 4

Solve for $m$ .

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

$m^{2} + 8 m + 16 = m + 10$

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

Subtract $m$ from both sides of the equation.

$m^{2} + 8 m + 16 - m = 10$

Subtract $m$ from $8 m$ .

$m^{2} + 7 m + 16 = 10$

Subtract $10$ from both sides of the equation.

$m^{2} + 7 m + 16 - 10 = 0$

Subtract $10$ from $16$ .

$m^{2} + 7 m + 6 = 0$

Factor $m^{2} + 7 m + 6$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $6$ and whose sum is $7$ .

$1, 6$

Write the factored form using these integers.

$(m + 1) (m + 6) = 0$

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

$m + 1 = 0$

$m + 6 = 0$

Set $m + 1$ equal to $0$ and solve for $m$ .

Set $m + 1$ equal to $0$ .

$m + 1 = 0$

Subtract $1$ from both sides of the equation.

$m = - 1$

Set $m + 6$ equal to $0$ and solve for $m$ .

Set $m + 6$ equal to $0$ .

$m + 6 = 0$

Subtract $6$ from both sides of the equation.

$m = - 6$

The final solution is all the values that make $(m + 1) (m + 6) = 0$ true.

$m = - 1, - 6$

Step 5

Exclude the solutions that do not make $m + 4 = \sqrt{m + 10}$ true.

$m = - 1$

Solve for m m+4 = square root of m+10

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