Algebra Examples

$\sqrt{y^{2} - 12 y + 36} = - 1$

Step 1

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

${\sqrt{y^{2} - 12 y + 36}}^{2} = {(- 1)}^{2}$

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 $\sqrt{y^{2} - 12 y + 36}$ as ${(y^{2} - 12 y + 36)}^{\frac{1}{2}}$ .

${({(y^{2} - 12 y + 36)}^{\frac{1}{2}})}^{2} = {(- 1)}^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(y^{2} - 12 y + 36)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(y^{2} - 12 y + 36)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

${(y^{2} - 12 y + 36)}^{\frac{1}{2} \cdot 2} = {(- 1)}^{2}$

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.

${(y^{2} - 12 y + 36)}^{\frac{1}{2} \cdot 2} = {(- 1)}^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(y^{2} - 12 y + 36)}^{1} = {(- 1)}^{2}$

Step 2.2.1.2

Simplify.

$y^{2} - 12 y + 36 = {(- 1)}^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

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

$y^{2} - 12 y + 36 = 1$

Step 3

Solve for $y$ .

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Step 3.1

Subtract $1$ from both sides of the equation.

$y^{2} - 12 y + 36 - 1 = 0$

Step 3.2

Subtract $1$ from $36$ .

$y^{2} - 12 y + 35 = 0$

Step 3.3

Factor $y^{2} - 12 y + 35$ using the AC method.

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Step 3.3.1

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 $35$ and whose sum is $- 12$ .

$- 7, - 5$

Step 3.3.2

Write the factored form using these integers.

$(y - 7) (y - 5) = 0$

Step 3.4

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

$y - 7 = 0$

$y - 5 = 0$

Step 3.5

Set $y - 7$ equal to $0$ and solve for $y$ .

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Step 3.5.1

Set $y - 7$ equal to $0$ .

$y - 7 = 0$

Step 3.5.2

Add $7$ to both sides of the equation.

$y = 7$

Step 3.6

Set $y - 5$ equal to $0$ and solve for $y$ .

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Step 3.6.1

Set $y - 5$ equal to $0$ .

$y - 5 = 0$

Step 3.6.2

Add $5$ to both sides of the equation.

$y = 5$

Step 3.7

The final solution is all the values that make $(y - 7) (y - 5) = 0$ true.

$y = 7, 5$

Step 4

Exclude the solutions that do not make $\sqrt{y^{2} - 12 y + 36} = - 1$ true.

$No solution$