Basic Math Examples

$6 a^{2} + 15 a = 4.5$

Step 1

Subtract $4.5$ from both sides of the equation.

$6 a^{2} + 15 a - 4.5 = 0$

Step 2

Factor $1.5$ out of $6 a^{2} + 15 a - 4.5$ .

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Step 2.1

Factor $1.5$ out of $6 a^{2}$ .

$1.5 (4 a^{2}) + 15 a - 4.5 = 0$

Step 2.2

Factor $1.5$ out of $15 a$ .

$1.5 (4 a^{2}) + 1.5 (10 a) - 4.5 = 0$

Step 2.3

Factor $1.5$ out of $- 4.5$ .

$1.5 (4 a^{2}) + 1.5 (10 a) + 1.5 (- 3) = 0$

Step 2.4

Factor $1.5$ out of $1.5 (4 a^{2}) + 1.5 (10 a)$ .

$1.5 (4 a^{2} + 10 a) + 1.5 (- 3) = 0$

Step 2.5

Factor $1.5$ out of $1.5 (4 a^{2} + 10 a) + 1.5 (- 3)$ .

$1.5 (4 a^{2} + 10 a - 3) = 0$

Step 3

Divide each term in $1.5 (4 a^{2} + 10 a - 3) = 0$ by $1.5$ and simplify.

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Step 3.1

Divide each term in $1.5 (4 a^{2} + 10 a - 3) = 0$ by $1.5$ .

$\frac{1.5 (4 a^{2} + 10 a - 3)}{1.5} = \frac{0}{1.5}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $1.5$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{1.5 (4 a^{2} + 10 a - 3)}{1.5} = \frac{0}{1.5}$

Step 3.2.1.2

Divide $4 a^{2} + 10 a - 3$ by $1$ .

$4 a^{2} + 10 a - 3 = \frac{0}{1.5}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $0$ by $1.5$ .

$4 a^{2} + 10 a - 3 = 0$

Step 4

Use the quadratic formula to find the solutions.

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

Step 5

Substitute the values $a = 4$ , $b = 10$ , and $c = - 3$ into the quadratic formula and solve for $a$ .

$\frac{- 10 \pm \sqrt{10^{2} - 4 \cdot (4 \cdot - 3)}}{2 \cdot 4}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $10$ to the power of $2$ .

$a = \frac{- 10 \pm \sqrt{100 - 4 \cdot 4 \cdot - 3}}{2 \cdot 4}$

Step 6.1.2

Multiply $- 4 \cdot 4 \cdot - 3$ .

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Step 6.1.2.1

Multiply $- 4$ by $4$ .

$a = \frac{- 10 \pm \sqrt{100 - 16 \cdot - 3}}{2 \cdot 4}$

Step 6.1.2.2

Multiply $- 16$ by $- 3$ .

$a = \frac{- 10 \pm \sqrt{100 + 48}}{2 \cdot 4}$

Step 6.1.3

Add $100$ and $48$ .

$a = \frac{- 10 \pm \sqrt{148}}{2 \cdot 4}$

Step 6.1.4

Rewrite $148$ as $2^{2} \cdot 37$ .

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Step 6.1.4.1

Factor $4$ out of $148$ .

$a = \frac{- 10 \pm \sqrt{4 (37)}}{2 \cdot 4}$

Step 6.1.4.2

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

$a = \frac{- 10 \pm \sqrt{2^{2} \cdot 37}}{2 \cdot 4}$

Step 6.1.5

Pull terms out from under the radical.

$a = \frac{- 10 \pm 2 \sqrt{37}}{2 \cdot 4}$

Step 6.2

Multiply $2$ by $4$ .

$a = \frac{- 10 \pm 2 \sqrt{37}}{8}$

Step 6.3

Simplify $\frac{- 10 \pm 2 \sqrt{37}}{8}$ .

$a = \frac{- 5 \pm \sqrt{37}}{4}$

Step 7

The final answer is the combination of both solutions.

$a = - \frac{5 - \sqrt{37}}{4}, - \frac{5 + \sqrt{37}}{4}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$a = - \frac{5 - \sqrt{37}}{4}, - \frac{5 + \sqrt{37}}{4}$

Decimal Form:

$a = 0.27069063 \dots, - 2.77069063 \dots$