Finite Math Examples

$a = \sqrt{7 a - 12}$

Step 1

Subtract $\sqrt{7 a - 12}$ from both sides of the equation.

$a - \sqrt{7 a - 12} = 0$

Step 2

Subtract $a$ from both sides of the equation.

$- \sqrt{7 a - 12} = - a$

Step 3

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

${(- \sqrt{7 a - 12})}^{2} = {(- a)}^{2}$

Step 4

Simplify each side of the equation.

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Step 4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7 a - 12}$ as ${(7 a - 12)}^{\frac{1}{2}}$ .

${(- {(7 a - 12)}^{\frac{1}{2}})}^{2} = {(- a)}^{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify ${(- {(7 a - 12)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.1.1

Apply the product rule to $- {(7 a - 12)}^{\frac{1}{2}}$ .

${(- 1)}^{2} {({(7 a - 12)}^{\frac{1}{2}})}^{2} = {(- a)}^{2}$

Step 4.2.1.2

Raise $- 1$ to the power of $2$ .

$1 {({(7 a - 12)}^{\frac{1}{2}})}^{2} = {(- a)}^{2}$

Step 4.2.1.3

Multiply ${({(7 a - 12)}^{\frac{1}{2}})}^{2}$ by $1$ .

${({(7 a - 12)}^{\frac{1}{2}})}^{2} = {(- a)}^{2}$

Step 4.2.1.4

Multiply the exponents in ${({(7 a - 12)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.1.4.1

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

${(7 a - 12)}^{\frac{1}{2} \cdot 2} = {(- a)}^{2}$

Step 4.2.1.4.2

Cancel the common factor of $2$ .

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Step 4.2.1.4.2.1

Cancel the common factor.

${(7 a - 12)}^{\frac{1}{2} \cdot 2} = {(- a)}^{2}$

Step 4.2.1.4.2.2

Rewrite the expression.

${(7 a - 12)}^{1} = {(- a)}^{2}$

Step 4.2.1.5

Simplify.

$7 a - 12 = {(- a)}^{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

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

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Step 4.3.1.1

Apply the product rule to $- a$ .

$7 a - 12 = {(- 1)}^{2} a^{2}$

Step 4.3.1.2

Raise $- 1$ to the power of $2$ .

$7 a - 12 = 1 a^{2}$

Step 4.3.1.3

Multiply $a^{2}$ by $1$ .

$7 a - 12 = a^{2}$

Step 5

Solve for $a$ .

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Step 5.1

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

$7 a - 12 - a^{2} = 0$

Step 5.2

Factor the left side of the equation.

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Step 5.2.1

Factor $- 1$ out of $7 a - 12 - a^{2}$ .

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Step 5.2.1.1

Reorder the expression.

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Step 5.2.1.1.1

Move $- 12$ .

$7 a - a^{2} - 12 = 0$

Step 5.2.1.1.2

Reorder $7 a$ and $- a^{2}$ .

$- a^{2} + 7 a - 12 = 0$

Step 5.2.1.2

Factor $- 1$ out of $- a^{2}$ .

$- (a^{2}) + 7 a - 12 = 0$

Step 5.2.1.3

Factor $- 1$ out of $7 a$ .

$- (a^{2}) - (- 7 a) - 12 = 0$

Step 5.2.1.4

Rewrite $- 12$ as $- 1 (12)$ .

$- (a^{2}) - (- 7 a) - 1 \cdot 12 = 0$

Step 5.2.1.5

Factor $- 1$ out of $- (a^{2}) - (- 7 a)$ .

$- (a^{2} - 7 a) - 1 \cdot 12 = 0$

Step 5.2.1.6

Factor $- 1$ out of $- (a^{2} - 7 a) - 1 (12)$ .

$- (a^{2} - 7 a + 12) = 0$

Step 5.2.2

Factor.

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Step 5.2.2.1

Factor $a^{2} - 7 a + 12$ using the AC method.

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Step 5.2.2.1.1

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 $12$ and whose sum is $- 7$ .

$- 4, - 3$

Step 5.2.2.1.2

Write the factored form using these integers.

$- ((a - 4) (a - 3)) = 0$

Step 5.2.2.2

Remove unnecessary parentheses.

$- (a - 4) (a - 3) = 0$

Step 5.3

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

$a - 4 = 0$

$a - 3 = 0$

Step 5.4

Set $a - 4$ equal to $0$ and solve for $a$ .

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Step 5.4.1

Set $a - 4$ equal to $0$ .

$a - 4 = 0$

Step 5.4.2

Add $4$ to both sides of the equation.

$a = 4$

Step 5.5

Set $a - 3$ equal to $0$ and solve for $a$ .

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Step 5.5.1

Set $a - 3$ equal to $0$ .

$a - 3 = 0$

Step 5.5.2

Add $3$ to both sides of the equation.

$a = 3$

Step 5.6

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

$a = 4, 3$