Finite Math Examples

$q^{2} + 8 q + 16 = - 9 q^{2} + 6 q + 426$

Step 1

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

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Step 1.1

Add $9 q^{2}$ to both sides of the equation.

$q^{2} + 8 q + 16 + 9 q^{2} = 6 q + 426$

Step 1.2

Subtract $6 q$ from both sides of the equation.

$q^{2} + 8 q + 16 + 9 q^{2} - 6 q = 426$

Step 1.3

Add $q^{2}$ and $9 q^{2}$ .

$10 q^{2} + 8 q + 16 - 6 q = 426$

Step 1.4

Subtract $6 q$ from $8 q$ .

$10 q^{2} + 2 q + 16 = 426$

Step 2

Subtract $426$ from both sides of the equation.

$10 q^{2} + 2 q + 16 - 426 = 0$

Step 3

Subtract $426$ from $16$ .

$10 q^{2} + 2 q - 410 = 0$

Step 4

Factor $2$ out of $10 q^{2} + 2 q - 410$ .

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Step 4.1

Factor $2$ out of $10 q^{2}$ .

$2 (5 q^{2}) + 2 q - 410 = 0$

Step 4.2

Factor $2$ out of $2 q$ .

$2 (5 q^{2}) + 2 (q) - 410 = 0$

Step 4.3

Factor $2$ out of $- 410$ .

$2 (5 q^{2}) + 2 q + 2 \cdot - 205 = 0$

Step 4.4

Factor $2$ out of $2 (5 q^{2}) + 2 q$ .

$2 (5 q^{2} + q) + 2 \cdot - 205 = 0$

Step 4.5

Factor $2$ out of $2 (5 q^{2} + q) + 2 \cdot - 205$ .

$2 (5 q^{2} + q - 205) = 0$

Step 5

Divide each term in $2 (5 q^{2} + q - 205) = 0$ by $2$ and simplify.

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Step 5.1

Divide each term in $2 (5 q^{2} + q - 205) = 0$ by $2$ .

$\frac{2 (5 q^{2} + q - 205)}{2} = \frac{0}{2}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $2$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{2 (5 q^{2} + q - 205)}{2} = \frac{0}{2}$

Step 5.2.1.2

Divide $5 q^{2} + q - 205$ by $1$ .

$5 q^{2} + q - 205 = \frac{0}{2}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Divide $0$ by $2$ .

$5 q^{2} + q - 205 = 0$

Step 6

Use the quadratic formula to find the solutions.

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

Step 7

Substitute the values $a = 5$ , $b = 1$ , and $c = - 205$ into the quadratic formula and solve for $q$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (5 \cdot - 205)}}{2 \cdot 5}$

Step 8

Simplify.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

One to any power is one.

$q = \frac{- 1 \pm \sqrt{1 - 4 \cdot 5 \cdot - 205}}{2 \cdot 5}$

Step 8.1.2

Multiply $- 4 \cdot 5 \cdot - 205$ .

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Step 8.1.2.1

Multiply $- 4$ by $5$ .

$q = \frac{- 1 \pm \sqrt{1 - 20 \cdot - 205}}{2 \cdot 5}$

Step 8.1.2.2

Multiply $- 20$ by $- 205$ .

$q = \frac{- 1 \pm \sqrt{1 + 4100}}{2 \cdot 5}$

Step 8.1.3

Add $1$ and $4100$ .

$q = \frac{- 1 \pm \sqrt{4101}}{2 \cdot 5}$

Step 8.2

Multiply $2$ by $5$ .

$q = \frac{- 1 \pm \sqrt{4101}}{10}$

Step 9

The final answer is the combination of both solutions.

$q = - \frac{1 - \sqrt{4101}}{10}, - \frac{1 + \sqrt{4101}}{10}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$q = - \frac{1 - \sqrt{4101}}{10}, - \frac{1 + \sqrt{4101}}{10}$

Decimal Form:

$q = 6.30390505 \dots, - 6.50390505 \dots$