Basic Math Examples

$(a + 2 (- a^{2} + 6 a - 8)) = 0$

Step 1

Simplify $a + 2 (- a^{2} + 6 a - 8)$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

$a + 2 (- a^{2}) + 2 (6 a) + 2 \cdot - 8 = 0$

Step 1.1.2

Simplify.

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Step 1.1.2.1

Multiply $- 1$ by $2$ .

$a - 2 a^{2} + 2 (6 a) + 2 \cdot - 8 = 0$

Step 1.1.2.2

Multiply $6$ by $2$ .

$a - 2 a^{2} + 12 a + 2 \cdot - 8 = 0$

Step 1.1.2.3

Multiply $2$ by $- 8$ .

$a - 2 a^{2} + 12 a - 16 = 0$

Step 1.2

Add $a$ and $12 a$ .

$- 2 a^{2} + 13 a - 16 = 0$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values $a = - 2$ , $b = 13$ , and $c = - 16$ into the quadratic formula and solve for $a$ .

$\frac{- 13 \pm \sqrt{13^{2} - 4 \cdot (- 2 \cdot - 16)}}{2 \cdot - 2}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $13$ to the power of $2$ .

$a = \frac{- 13 \pm \sqrt{169 - 4 \cdot - 2 \cdot - 16}}{2 \cdot - 2}$

Step 4.1.2

Multiply $- 4 \cdot - 2 \cdot - 16$ .

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Step 4.1.2.1

Multiply $- 4$ by $- 2$ .

$a = \frac{- 13 \pm \sqrt{169 + 8 \cdot - 16}}{2 \cdot - 2}$

Step 4.1.2.2

Multiply $8$ by $- 16$ .

$a = \frac{- 13 \pm \sqrt{169 - 128}}{2 \cdot - 2}$

Step 4.1.3

Subtract $128$ from $169$ .

$a = \frac{- 13 \pm \sqrt{41}}{2 \cdot - 2}$

Step 4.2

Multiply $2$ by $- 2$ .

$a = \frac{- 13 \pm \sqrt{41}}{- 4}$

Step 4.3

Simplify $\frac{- 13 \pm \sqrt{41}}{- 4}$ .

$a = \frac{13 \pm \sqrt{41}}{4}$

Step 5

The final answer is the combination of both solutions.

$a = \frac{13 + \sqrt{41}}{4}, \frac{13 - \sqrt{41}}{4}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$a = \frac{13 + \sqrt{41}}{4}, \frac{13 - \sqrt{41}}{4}$

Decimal Form:

$a = 4.85078105 \dots, 1.64921894 \dots$