Basic Math Examples

$\sqrt{y - 17} + 14 = y$

Step 1

Subtract $14$ from both sides of the equation.

$\sqrt{y - 17} = y - 14$

Step 2

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

${\sqrt{y - 17}}^{2} = {(y - 14)}^{2}$

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 $\sqrt{y - 17}$ as ${(y - 17)}^{\frac{1}{2}}$ .

${({(y - 17)}^{\frac{1}{2}})}^{2} = {(y - 14)}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(y - 17)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(y - 17)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

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

${(y - 17)}^{\frac{1}{2} \cdot 2} = {(y - 14)}^{2}$

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.

${(y - 17)}^{\frac{1}{2} \cdot 2} = {(y - 14)}^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(y - 17)}^{1} = {(y - 14)}^{2}$

Step 3.2.1.2

Simplify.

$y - 17 = {(y - 14)}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${(y - 14)}^{2}$ .

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Step 3.3.1.1

Rewrite ${(y - 14)}^{2}$ as $(y - 14) (y - 14)$ .

$y - 17 = (y - 14) (y - 14)$

Step 3.3.1.2

Expand $(y - 14) (y - 14)$ using the FOIL Method.

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Step 3.3.1.2.1

Apply the distributive property.

$y - 17 = y (y - 14) - 14 (y - 14)$

Step 3.3.1.2.2

Apply the distributive property.

$y - 17 = y \cdot y + y \cdot - 14 - 14 (y - 14)$

Step 3.3.1.2.3

Apply the distributive property.

$y - 17 = y \cdot y + y \cdot - 14 - 14 y - 14 \cdot - 14$

Step 3.3.1.3

Simplify and combine like terms.

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Step 3.3.1.3.1

Simplify each term.

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Step 3.3.1.3.1.1

Multiply $y$ by $y$ .

$y - 17 = y^{2} + y \cdot - 14 - 14 y - 14 \cdot - 14$

Step 3.3.1.3.1.2

Move $- 14$ to the left of $y$ .

$y - 17 = y^{2} - 14 \cdot y - 14 y - 14 \cdot - 14$

Step 3.3.1.3.1.3

Multiply $- 14$ by $- 14$ .

$y - 17 = y^{2} - 14 y - 14 y + 196$

Step 3.3.1.3.2

Subtract $14 y$ from $- 14 y$ .

$y - 17 = y^{2} - 28 y + 196$

Step 4

Solve for $y$ .

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Step 4.1

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

$y^{2} - 28 y + 196 = y - 17$

Step 4.2

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

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Step 4.2.1

Subtract $y$ from both sides of the equation.

$y^{2} - 28 y + 196 - y = - 17$

Step 4.2.2

Subtract $y$ from $- 28 y$ .

$y^{2} - 29 y + 196 = - 17$

Step 4.3

Move all terms to the left side of the equation and simplify.

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Step 4.3.1

Add $17$ to both sides of the equation.

$y^{2} - 29 y + 196 + 17 = 0$

Step 4.3.2

Add $196$ and $17$ .

$y^{2} - 29 y + 213 = 0$

Step 4.4

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4.5

Substitute the values $a = 1$ , $b = - 29$ , and $c = 213$ into the quadratic formula and solve for $y$ .

$\frac{29 \pm \sqrt{{(- 29)}^{2} - 4 \cdot (1 \cdot 213)}}{2 \cdot 1}$

Step 4.6

Simplify.

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Step 4.6.1

Simplify the numerator.

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Step 4.6.1.1

Raise $- 29$ to the power of $2$ .

$y = \frac{29 \pm \sqrt{841 - 4 \cdot 1 \cdot 213}}{2 \cdot 1}$

Step 4.6.1.2

Multiply $- 4 \cdot 1 \cdot 213$ .

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Step 4.6.1.2.1

Multiply $- 4$ by $1$ .

$y = \frac{29 \pm \sqrt{841 - 4 \cdot 213}}{2 \cdot 1}$

Step 4.6.1.2.2

Multiply $- 4$ by $213$ .

$y = \frac{29 \pm \sqrt{841 - 852}}{2 \cdot 1}$

Step 4.6.1.3

Subtract $852$ from $841$ .

$y = \frac{29 \pm \sqrt{- 11}}{2 \cdot 1}$

Step 4.6.1.4

Rewrite $- 11$ as $- 1 (11)$ .

$y = \frac{29 \pm \sqrt{- 1 \cdot 11}}{2 \cdot 1}$

Step 4.6.1.5

Rewrite $\sqrt{- 1 (11)}$ as $\sqrt{- 1} \cdot \sqrt{11}$ .

$y = \frac{29 \pm \sqrt{- 1} \cdot \sqrt{11}}{2 \cdot 1}$

Step 4.6.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$y = \frac{29 \pm i \sqrt{11}}{2 \cdot 1}$

Step 4.6.2

Multiply $2$ by $1$ .

$y = \frac{29 \pm i \sqrt{11}}{2}$

Step 4.7

The final answer is the combination of both solutions.

$y = \frac{29 + i \sqrt{11}}{2}, \frac{29 - i \sqrt{11}}{2}$