Finite Math Examples

$4 a^{2} + 8 a - 96 = 4$

Step 1

Subtract $4$ from both sides of the equation.

$4 a^{2} + 8 a - 96 - 4 = 0$

Step 2

Subtract $4$ from $- 96$ .

$4 a^{2} + 8 a - 100 = 0$

Step 3

Factor $4$ out of $4 a^{2} + 8 a - 100$ .

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Step 3.1

Factor $4$ out of $4 a^{2}$ .

$4 (a^{2}) + 8 a - 100 = 0$

Step 3.2

Factor $4$ out of $8 a$ .

$4 (a^{2}) + 4 (2 a) - 100 = 0$

Step 3.3

Factor $4$ out of $- 100$ .

$4 a^{2} + 4 (2 a) + 4 \cdot - 25 = 0$

Step 3.4

Factor $4$ out of $4 a^{2} + 4 (2 a)$ .

$4 (a^{2} + 2 a) + 4 \cdot - 25 = 0$

Step 3.5

Factor $4$ out of $4 (a^{2} + 2 a) + 4 \cdot - 25$ .

$4 (a^{2} + 2 a - 25) = 0$

Step 4

Divide each term in $4 (a^{2} + 2 a - 25) = 0$ by $4$ and simplify.

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Step 4.1

Divide each term in $4 (a^{2} + 2 a - 25) = 0$ by $4$ .

$\frac{4 (a^{2} + 2 a - 25)}{4} = \frac{0}{4}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $4$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{4 (a^{2} + 2 a - 25)}{4} = \frac{0}{4}$

Step 4.2.1.2

Divide $a^{2} + 2 a - 25$ by $1$ .

$a^{2} + 2 a - 25 = \frac{0}{4}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $0$ by $4$ .

$a^{2} + 2 a - 25 = 0$

Step 5

Use the quadratic formula to find the solutions.

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

Step 6

Substitute the values $a = 1$ , $b = 2$ , and $c = - 25$ into the quadratic formula and solve for $a$ .

$\frac{- 2 \pm \sqrt{2^{2} - 4 \cdot (1 \cdot - 25)}}{2 \cdot 1}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

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

$a = \frac{- 2 \pm \sqrt{4 - 4 \cdot 1 \cdot - 25}}{2 \cdot 1}$

Step 7.1.2

Multiply $- 4 \cdot 1 \cdot - 25$ .

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Step 7.1.2.1

Multiply $- 4$ by $1$ .

$a = \frac{- 2 \pm \sqrt{4 - 4 \cdot - 25}}{2 \cdot 1}$

Step 7.1.2.2

Multiply $- 4$ by $- 25$ .

$a = \frac{- 2 \pm \sqrt{4 + 100}}{2 \cdot 1}$

Step 7.1.3

Add $4$ and $100$ .

$a = \frac{- 2 \pm \sqrt{104}}{2 \cdot 1}$

Step 7.1.4

Rewrite $104$ as $2^{2} \cdot 26$ .

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Step 7.1.4.1

Factor $4$ out of $104$ .

$a = \frac{- 2 \pm \sqrt{4 (26)}}{2 \cdot 1}$

Step 7.1.4.2

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

$a = \frac{- 2 \pm \sqrt{2^{2} \cdot 26}}{2 \cdot 1}$

Step 7.1.5

Pull terms out from under the radical.

$a = \frac{- 2 \pm 2 \sqrt{26}}{2 \cdot 1}$

Step 7.2

Multiply $2$ by $1$ .

$a = \frac{- 2 \pm 2 \sqrt{26}}{2}$

Step 7.3

Simplify $\frac{- 2 \pm 2 \sqrt{26}}{2}$ .

$a = - 1 \pm \sqrt{26}$

Step 8

The final answer is the combination of both solutions.

$a = - 1 + \sqrt{26}, - 1 - \sqrt{26}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$a = - 1 + \sqrt{26}, - 1 - \sqrt{26}$

Decimal Form:

$a = 4.09901951 \dots, - 6.09901951 \dots$