Basic Math Examples

$5 q^{2} - 3 q = 4 a^{2} - 5 q + 15$

Step 1

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

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Step 1.1

Add $5 q$ to both sides of the equation.

$5 q^{2} - 3 q + 5 q = 4 a^{2} + 15$

Step 1.2

Add $- 3 q$ and $5 q$ .

$5 q^{2} + 2 q = 4 a^{2} + 15$

Step 2

Move all the expressions to the left side of the equation.

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Step 2.1

Subtract $4 a^{2}$ from both sides of the equation.

$5 q^{2} + 2 q - 4 a^{2} = 15$

Step 2.2

Subtract $15$ from both sides of the equation.

$5 q^{2} + 2 q - 4 a^{2} - 15 = 0$

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = 5$ , $b = 2$ , and $c = - 4 a^{2} - 15$ into the quadratic formula and solve for $q$ .

$\frac{- 2 \pm \sqrt{2^{2} - 4 \cdot (5 \cdot (- 4 a^{2} - 15))}}{2 \cdot 5}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

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

$q = \frac{- 2 \pm \sqrt{4 - 4 \cdot 5 \cdot (- 4 a^{2} - 15)}}{2 \cdot 5}$

Step 5.1.2

Multiply $- 4$ by $5$ .

$q = \frac{- 2 \pm \sqrt{4 - 20 \cdot (- 4 a^{2} - 15)}}{2 \cdot 5}$

Step 5.1.3

Apply the distributive property.

$q = \frac{- 2 \pm \sqrt{4 - 20 (- 4 a^{2}) - 20 \cdot - 15}}{2 \cdot 5}$

Step 5.1.4

Multiply $- 4$ by $- 20$ .

$q = \frac{- 2 \pm \sqrt{4 + 80 a^{2} - 20 \cdot - 15}}{2 \cdot 5}$

Step 5.1.5

Multiply $- 20$ by $- 15$ .

$q = \frac{- 2 \pm \sqrt{4 + 80 a^{2} + 300}}{2 \cdot 5}$

Step 5.1.6

Add $4$ and $300$ .

$q = \frac{- 2 \pm \sqrt{80 a^{2} + 304}}{2 \cdot 5}$

Step 5.1.7

Factor $16$ out of $80 a^{2} + 304$ .

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Step 5.1.7.1

Factor $16$ out of $80 a^{2}$ .

$q = \frac{- 2 \pm \sqrt{16 (5 a^{2}) + 304}}{2 \cdot 5}$

Step 5.1.7.2

Factor $16$ out of $304$ .

$q = \frac{- 2 \pm \sqrt{16 (5 a^{2}) + 16 (19)}}{2 \cdot 5}$

Step 5.1.7.3

Factor $16$ out of $16 (5 a^{2}) + 16 (19)$ .

$q = \frac{- 2 \pm \sqrt{16 (5 a^{2} + 19)}}{2 \cdot 5}$

Step 5.1.8

Rewrite $16 (5 a^{2} + 19)$ as ${(2^{2})}^{2} (5 a^{2} + 19)$ .

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Step 5.1.8.1

Rewrite $16$ as $4^{2}$ .

$q = \frac{- 2 \pm \sqrt{4^{2} (5 a^{2} + 19)}}{2 \cdot 5}$

Step 5.1.8.2

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

$q = \frac{- 2 \pm \sqrt{{(2^{2})}^{2} (5 a^{2} + 19)}}{2 \cdot 5}$

Step 5.1.9

Pull terms out from under the radical.

$q = \frac{- 2 \pm 2^{2} \sqrt{5 a^{2} + 19}}{2 \cdot 5}$

Step 5.1.10

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

$q = \frac{- 2 \pm 4 \sqrt{5 a^{2} + 19}}{2 \cdot 5}$

Step 5.2

Multiply $2$ by $5$ .

$q = \frac{- 2 \pm 4 \sqrt{5 a^{2} + 19}}{10}$

Step 5.3

Simplify $\frac{- 2 \pm 4 \sqrt{5 a^{2} + 19}}{10}$ .

$q = \frac{- 1 \pm 2 \sqrt{5 a^{2} + 19}}{5}$

Step 6

The final answer is the combination of both solutions.

$q = - \frac{1 - 2 \sqrt{5 a^{2} + 19}}{5}$

$q = - \frac{1 + 2 \sqrt{5 a^{2} + 19}}{5}$