Algebra Examples

$\sqrt{7 y + 18} = y$

Step 1

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

${\sqrt{7 y + 18}}^{2} = y^{2}$

Step 2

Simplify each side of the equation.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7 y + 18}$ as ${(7 y + 18)}^{\frac{1}{2}}$ .

${({(7 y + 18)}^{\frac{1}{2}})}^{2} = y^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(7 y + 18)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(7 y + 18)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

${(7 y + 18)}^{\frac{1}{2} \cdot 2} = y^{2}$

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

${(7 y + 18)}^{\frac{1}{2} \cdot 2} = y^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(7 y + 18)}^{1} = y^{2}$

Step 2.2.1.2

Simplify.

$7 y + 18 = y^{2}$

Step 3

Solve for $y$ .

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Step 3.1

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

$7 y + 18 - y^{2} = 0$

Step 3.2

Factor the left side of the equation.

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Step 3.2.1

Factor $- 1$ out of $7 y + 18 - y^{2}$ .

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Step 3.2.1.1

Reorder the expression.

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Step 3.2.1.1.1

Move $18$ .

$7 y - y^{2} + 18 = 0$

Step 3.2.1.1.2

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

$- y^{2} + 7 y + 18 = 0$

Step 3.2.1.2

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

$- (y^{2}) + 7 y + 18 = 0$

Step 3.2.1.3

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

$- (y^{2}) - (- 7 y) + 18 = 0$

Step 3.2.1.4

Rewrite $18$ as $- 1 (- 18)$ .

$- (y^{2}) - (- 7 y) - 1 \cdot - 18 = 0$

Step 3.2.1.5

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

$- (y^{2} - 7 y) - 1 \cdot - 18 = 0$

Step 3.2.1.6

Factor $- 1$ out of $- (y^{2} - 7 y) - 1 (- 18)$ .

$- (y^{2} - 7 y - 18) = 0$

Step 3.2.2

Factor.

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Step 3.2.2.1

Factor $y^{2} - 7 y - 18$ using the AC method.

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Step 3.2.2.1.1

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 $- 18$ and whose sum is $- 7$ .

$- 9, 2$

Step 3.2.2.1.2

Write the factored form using these integers.

$- ((y - 9) (y + 2)) = 0$

Step 3.2.2.2

Remove unnecessary parentheses.

$- (y - 9) (y + 2) = 0$

Step 3.3

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

$y - 9 = 0$

$y + 2 = 0$

Step 3.4

Set $y - 9$ equal to $0$ and solve for $y$ .

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Step 3.4.1

Set $y - 9$ equal to $0$ .

$y - 9 = 0$

Step 3.4.2

Add $9$ to both sides of the equation.

$y = 9$

Step 3.5

Set $y + 2$ equal to $0$ and solve for $y$ .

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Step 3.5.1

Set $y + 2$ equal to $0$ .

$y + 2 = 0$

Step 3.5.2

Subtract $2$ from both sides of the equation.

$y = - 2$

Step 3.6

The final solution is all the values that make $- (y - 9) (y + 2) = 0$ true.

$y = 9, - 2$

Step 4

Exclude the solutions that do not make $\sqrt{7 y + 18} = y$ true.

$y = 9$