Basic Math Examples

$45 y^{2} - 48 y - 20 = 0$

Step 1

Use the quadratic formula to find the solutions.

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

Step 2

Substitute the values $a = 45$ , $b = - 48$ , and $c = - 20$ into the quadratic formula and solve for $y$ .

$\frac{48 \pm \sqrt{{(- 48)}^{2} - 4 \cdot (45 \cdot - 20)}}{2 \cdot 45}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise $- 48$ to the power of $2$ .

$y = \frac{48 \pm \sqrt{2304 - 4 \cdot 45 \cdot - 20}}{2 \cdot 45}$

Step 3.1.2

Multiply $- 4 \cdot 45 \cdot - 20$ .

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Step 3.1.2.1

Multiply $- 4$ by $45$ .

$y = \frac{48 \pm \sqrt{2304 - 180 \cdot - 20}}{2 \cdot 45}$

Step 3.1.2.2

Multiply $- 180$ by $- 20$ .

$y = \frac{48 \pm \sqrt{2304 + 3600}}{2 \cdot 45}$

Step 3.1.3

Add $2304$ and $3600$ .

$y = \frac{48 \pm \sqrt{5904}}{2 \cdot 45}$

Step 3.1.4

Rewrite $5904$ as $12^{2} \cdot 41$ .

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Step 3.1.4.1

Factor $144$ out of $5904$ .

$y = \frac{48 \pm \sqrt{144 (41)}}{2 \cdot 45}$

Step 3.1.4.2

Rewrite $144$ as $12^{2}$ .

$y = \frac{48 \pm \sqrt{12^{2} \cdot 41}}{2 \cdot 45}$

Step 3.1.5

Pull terms out from under the radical.

$y = \frac{48 \pm 12 \sqrt{41}}{2 \cdot 45}$

Step 3.2

Multiply $2$ by $45$ .

$y = \frac{48 \pm 12 \sqrt{41}}{90}$

Step 3.3

Simplify $\frac{48 \pm 12 \sqrt{41}}{90}$ .

$y = \frac{8 \pm 2 \sqrt{41}}{15}$

Step 4

The final answer is the combination of both solutions.

$y = \frac{8 + 2 \sqrt{41}}{15}, \frac{8 - 2 \sqrt{41}}{15}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$y = \frac{8 + 2 \sqrt{41}}{15}, \frac{8 - 2 \sqrt{41}}{15}$

Decimal Form:

$y = 1.38708323 \dots, - 0.32041656 \dots$