Basic Math Examples

$(y - 5^{2}) = y (y - 8) + 11$

Step 1

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

$y (y - 8) + 11 = y - 5^{2}$

Step 2

Simplify each term.

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Step 2.1

Apply the distributive property.

$y \cdot y + y \cdot - 8 + 11 = y - 5^{2}$

Step 2.2

Multiply $y$ by $y$ .

$y^{2} + y \cdot - 8 + 11 = y - 5^{2}$

Step 2.3

Move $- 8$ to the left of $y$ .

$y^{2} - 8 y + 11 = y - 5^{2}$

Step 3

Simplify each term.

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Step 3.1

Raise $5$ to the power of $2$ .

$y^{2} - 8 y + 11 = y - 1 \cdot 25$

Step 3.2

Multiply $- 1$ by $25$ .

$y^{2} - 8 y + 11 = y - 25$

Step 4

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

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Step 4.1

Subtract $y$ from both sides of the equation.

$y^{2} - 8 y + 11 - y = - 25$

Step 4.2

Subtract $y$ from $- 8 y$ .

$y^{2} - 9 y + 11 = - 25$

Step 5

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

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Step 5.1

Add $25$ to both sides of the equation.

$y^{2} - 9 y + 11 + 25 = 0$

Step 5.2

Add $11$ and $25$ .

$y^{2} - 9 y + 36 = 0$

Step 6

Use the quadratic formula to find the solutions.

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

Step 7

Substitute the values $a = 1$ , $b = - 9$ , and $c = 36$ into the quadratic formula and solve for $y$ .

$\frac{9 \pm \sqrt{{(- 9)}^{2} - 4 \cdot (1 \cdot 36)}}{2 \cdot 1}$

Step 8

Simplify.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raise $- 9$ to the power of $2$ .

$y = \frac{9 \pm \sqrt{81 - 4 \cdot 1 \cdot 36}}{2 \cdot 1}$

Step 8.1.2

Multiply $- 4 \cdot 1 \cdot 36$ .

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Step 8.1.2.1

Multiply $- 4$ by $1$ .

$y = \frac{9 \pm \sqrt{81 - 4 \cdot 36}}{2 \cdot 1}$

Step 8.1.2.2

Multiply $- 4$ by $36$ .

$y = \frac{9 \pm \sqrt{81 - 144}}{2 \cdot 1}$

Step 8.1.3

Subtract $144$ from $81$ .

$y = \frac{9 \pm \sqrt{- 63}}{2 \cdot 1}$

Step 8.1.4

Rewrite $- 63$ as $- 1 (63)$ .

$y = \frac{9 \pm \sqrt{- 1 \cdot 63}}{2 \cdot 1}$

Step 8.1.5

Rewrite $\sqrt{- 1 (63)}$ as $\sqrt{- 1} \cdot \sqrt{63}$ .

$y = \frac{9 \pm \sqrt{- 1} \cdot \sqrt{63}}{2 \cdot 1}$

Step 8.1.6

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

$y = \frac{9 \pm i \cdot \sqrt{63}}{2 \cdot 1}$

Step 8.1.7

Rewrite $63$ as $3^{2} \cdot 7$ .

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Step 8.1.7.1

Factor $9$ out of $63$ .

$y = \frac{9 \pm i \cdot \sqrt{9 (7)}}{2 \cdot 1}$

Step 8.1.7.2

Rewrite $9$ as $3^{2}$ .

$y = \frac{9 \pm i \cdot \sqrt{3^{2} \cdot 7}}{2 \cdot 1}$

Step 8.1.8

Pull terms out from under the radical.

$y = \frac{9 \pm i \cdot (3 \sqrt{7})}{2 \cdot 1}$

Step 8.1.9

Move $3$ to the left of $i$ .

$y = \frac{9 \pm 3 i \sqrt{7}}{2 \cdot 1}$

Step 8.2

Multiply $2$ by $1$ .

$y = \frac{9 \pm 3 i \sqrt{7}}{2}$

Step 9

The final answer is the combination of both solutions.

$y = \frac{9 + 3 i \sqrt{7}}{2}, \frac{9 - 3 i \sqrt{7}}{2}$