Basic Math Examples

$2 y^{2} + 24 y + 12 = y^{2} + 32 - 3$

Step 1

Subtract $3$ from $32$ .

$2 y^{2} + 24 y + 12 = y^{2} + 29$

Step 2

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

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Step 2.1

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

$2 y^{2} + 24 y + 12 - y^{2} = 29$

Step 2.2

Subtract $y^{2}$ from $2 y^{2}$ .

$y^{2} + 24 y + 12 = 29$

Step 3

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

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Step 3.1

Subtract $29$ from both sides of the equation.

$y^{2} + 24 y + 12 - 29 = 0$

Step 3.2

Subtract $29$ from $12$ .

$y^{2} + 24 y - 17 = 0$

Step 4

Use the quadratic formula to find the solutions.

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

Step 5

Substitute the values $a = 1$ , $b = 24$ , and $c = - 17$ into the quadratic formula and solve for $y$ .

$\frac{- 24 \pm \sqrt{24^{2} - 4 \cdot (1 \cdot - 17)}}{2 \cdot 1}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $24$ to the power of $2$ .

$y = \frac{- 24 \pm \sqrt{576 - 4 \cdot 1 \cdot - 17}}{2 \cdot 1}$

Step 6.1.2

Multiply $- 4 \cdot 1 \cdot - 17$ .

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Step 6.1.2.1

Multiply $- 4$ by $1$ .

$y = \frac{- 24 \pm \sqrt{576 - 4 \cdot - 17}}{2 \cdot 1}$

Step 6.1.2.2

Multiply $- 4$ by $- 17$ .

$y = \frac{- 24 \pm \sqrt{576 + 68}}{2 \cdot 1}$

Step 6.1.3

Add $576$ and $68$ .

$y = \frac{- 24 \pm \sqrt{644}}{2 \cdot 1}$

Step 6.1.4

Rewrite $644$ as $2^{2} \cdot 161$ .

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Step 6.1.4.1

Factor $4$ out of $644$ .

$y = \frac{- 24 \pm \sqrt{4 (161)}}{2 \cdot 1}$

Step 6.1.4.2

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

$y = \frac{- 24 \pm \sqrt{2^{2} \cdot 161}}{2 \cdot 1}$

Step 6.1.5

Pull terms out from under the radical.

$y = \frac{- 24 \pm 2 \sqrt{161}}{2 \cdot 1}$

Step 6.2

Multiply $2$ by $1$ .

$y = \frac{- 24 \pm 2 \sqrt{161}}{2}$

Step 6.3

Simplify $\frac{- 24 \pm 2 \sqrt{161}}{2}$ .

$y = - 12 \pm \sqrt{161}$

Step 7

The final answer is the combination of both solutions.

$y = - 12 + \sqrt{161}, - 12 - \sqrt{161}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$y = - 12 + \sqrt{161}, - 12 - \sqrt{161}$

Decimal Form:

$y = 0.68857754 \dots, - 24.68857754 \dots$