Algebra Examples

$m = \sqrt{56 - m}$

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.

$\sqrt{56 - m} = m$

Step 2

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

${\sqrt{56 - m}}^{2} = m^{2}$

Step 3

Simplify each side of the equation.

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Step 3.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{56 - m}$ as

${(56 - m)}^{\frac{1}{2}}$ .

${({(56 - m)}^{\frac{1}{2}})}^{2} = m^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify

${({(56 - m)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in

${({(56 - m)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

${(56 - m)}^{\frac{1}{2} \cdot 2} = m^{2}$

Step 3.2.1.1.2

Cancel the common factor of

$2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

${(56 - m)}^{\frac{1}{2} \cdot 2} = m^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(56 - m)}^{1} = m^{2}$

Step 3.2.1.2

Simplify.

$56 - m = m^{2}$

Step 4

Solve for

$m$ .

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Step 4.1

Subtract

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

$56 - m - m^{2} = 0$

Step 4.2

Factor the left side of the equation.

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Step 4.2.1

Factor

$- 1$ out of

$56 - m - m^{2}$ .

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Step 4.2.1.1

Reorder the expression.

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Step 4.2.1.1.1

Move

$56$ .

$- m - m^{2} + 56 = 0$

Step 4.2.1.1.2

Reorder

$- m$ and

$- m^{2}$ .

$- m^{2} - m + 56 = 0$

Step 4.2.1.2

Factor

$- 1$ out of

$- m^{2}$ .

$- (m^{2}) - m + 56 = 0$

Step 4.2.1.3

Factor

$- 1$ out of

$- m$ .

$- (m^{2}) - (m) + 56 = 0$

Step 4.2.1.4

Rewrite

$56$ as

$- 1 (- 56)$ .

$- (m^{2}) - (m) - 1 \cdot - 56 = 0$

Step 4.2.1.5

Factor

$- 1$ out of

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

$- (m^{2} + m) - 1 \cdot - 56 = 0$

Step 4.2.1.6

Factor

$- 1$ out of

$- (m^{2} + m) - 1 (- 56)$ .

$- (m^{2} + m - 56) = 0$

Step 4.2.2

Factor.

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Step 4.2.2.1

Factor

$m^{2} + m - 56$ using the AC method.

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Step 4.2.2.1.1

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

$- 56$ and whose sum is

$1$ .

$- 7, 8$

Step 4.2.2.1.2

Write the factored form using these integers.

$- ((m - 7) (m + 8)) = 0$

Step 4.2.2.2

Remove unnecessary parentheses.

$- (m - 7) (m + 8) = 0$

Step 4.3

If any individual factor on the left side of the equation is equal to

$0$ , the entire expression will be equal to

$0$ .

$m - 7 = 0$

$m + 8 = 0$

Step 4.4

Set

$m - 7$ equal to

$0$ and solve for

$m$ .

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Step 4.4.1

Set

$m - 7$ equal to

$0$ .

$m - 7 = 0$

Step 4.4.2

Add

$7$ to both sides of the equation.

$m = 7$

Step 4.5

Set

$m + 8$ equal to

$0$ and solve for

$m$ .

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Step 4.5.1

Set

$m + 8$ equal to

$0$ .

$m + 8 = 0$

Step 4.5.2

Subtract

$8$ from both sides of the equation.

$m = - 8$

Step 4.6

The final solution is all the values that make

$- (m - 7) (m + 8) = 0$ true.

$m = 7, - 8$

Step 5

Exclude the solutions that do not make

$m = \sqrt{56 - m}$ true.

$m = 7$