Basic Math Examples

$(y - 3) (y + 3) + 23 = 8 y$

Step 1

Simplify $(y - 3) (y + 3) + 23$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Expand $(y - 3) (y + 3)$ using the FOIL Method.

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Step 1.1.1.1

Apply the distributive property.

$y (y + 3) - 3 (y + 3) + 23 = 8 y$

Step 1.1.1.2

Apply the distributive property.

$y \cdot y + y \cdot 3 - 3 (y + 3) + 23 = 8 y$

Step 1.1.1.3

Apply the distributive property.

$y \cdot y + y \cdot 3 - 3 y - 3 \cdot 3 + 23 = 8 y$

Step 1.1.2

Combine the opposite terms in $y \cdot y + y \cdot 3 - 3 y - 3 \cdot 3$ .

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Step 1.1.2.1

Reorder the factors in the terms $y \cdot 3$ and $- 3 y$ .

$y \cdot y + 3 y - 3 y - 3 \cdot 3 + 23 = 8 y$

Step 1.1.2.2

Subtract $3 y$ from $3 y$ .

$y \cdot y + 0 - 3 \cdot 3 + 23 = 8 y$

Step 1.1.2.3

Add $y \cdot y$ and $0$ .

$y \cdot y - 3 \cdot 3 + 23 = 8 y$

Step 1.1.3

Simplify each term.

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Step 1.1.3.1

Multiply $y$ by $y$ .

$y^{2} - 3 \cdot 3 + 23 = 8 y$

Step 1.1.3.2

Multiply $- 3$ by $3$ .

$y^{2} - 9 + 23 = 8 y$

Step 1.2

Add $- 9$ and $23$ .

$y^{2} + 14 = 8 y$

Step 2

Subtract $8 y$ from both sides of the equation.

$y^{2} + 14 - 8 y = 0$

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = 1$ , $b = - 8$ , and $c = 14$ into the quadratic formula and solve for $y$ .

$\frac{8 \pm \sqrt{{(- 8)}^{2} - 4 \cdot (1 \cdot 14)}}{2 \cdot 1}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $- 8$ to the power of $2$ .

$y = \frac{8 \pm \sqrt{64 - 4 \cdot 1 \cdot 14}}{2 \cdot 1}$

Step 5.1.2

Multiply $- 4 \cdot 1 \cdot 14$ .

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Step 5.1.2.1

Multiply $- 4$ by $1$ .

$y = \frac{8 \pm \sqrt{64 - 4 \cdot 14}}{2 \cdot 1}$

Step 5.1.2.2

Multiply $- 4$ by $14$ .

$y = \frac{8 \pm \sqrt{64 - 56}}{2 \cdot 1}$

Step 5.1.3

Subtract $56$ from $64$ .

$y = \frac{8 \pm \sqrt{8}}{2 \cdot 1}$

Step 5.1.4

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 5.1.4.1

Factor $4$ out of $8$ .

$y = \frac{8 \pm \sqrt{4 (2)}}{2 \cdot 1}$

Step 5.1.4.2

Rewrite $4$ as $2^{2}$ .

$y = \frac{8 \pm \sqrt{2^{2} \cdot 2}}{2 \cdot 1}$

Step 5.1.5

Pull terms out from under the radical.

$y = \frac{8 \pm 2 \sqrt{2}}{2 \cdot 1}$

Step 5.2

Multiply $2$ by $1$ .

$y = \frac{8 \pm 2 \sqrt{2}}{2}$

Step 5.3

Simplify $\frac{8 \pm 2 \sqrt{2}}{2}$ .

$y = 4 \pm \sqrt{2}$

Step 6

The final answer is the combination of both solutions.

$y = 4 + \sqrt{2}, 4 - \sqrt{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$y = 4 + \sqrt{2}, 4 - \sqrt{2}$

Decimal Form:

$y = 5.41421356 \dots, 2.58578643 \dots$