Basic Math Examples

$9 c + c (2 c - 3) = 9$

Step 1

Simplify $9 c + c (2 c - 3)$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

$9 c + c (2 c) + c \cdot - 3 = 9$

Step 1.1.2

Rewrite using the commutative property of multiplication.

$9 c + 2 c \cdot c + c \cdot - 3 = 9$

Step 1.1.3

Move $- 3$ to the left of $c$ .

$9 c + 2 c \cdot c - 3 \cdot c = 9$

Step 1.1.4

Multiply $c$ by $c$ by adding the exponents.

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Step 1.1.4.1

Move $c$ .

$9 c + 2 (c \cdot c) - 3 \cdot c = 9$

Step 1.1.4.2

Multiply $c$ by $c$ .

$9 c + 2 c^{2} - 3 \cdot c = 9$

$9 c + 2 c^{2} - 3 c = 9$

Step 1.2

Subtract $3 c$ from $9 c$ .

$2 c^{2} + 6 c = 9$

Step 2

Subtract $9$ from both sides of the equation.

$2 c^{2} + 6 c - 9 = 0$

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 $a = 2$ , $b = 6$ , and $c = - 9$ into the quadratic formula and solve for $c$ .

$\frac{- 6 \pm \sqrt{6^{2} - 4 \cdot (2 \cdot - 9)}}{2 \cdot 2}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $6$ to the power of $2$ .

$c = \frac{- 6 \pm \sqrt{36 - 4 \cdot 2 \cdot - 9}}{2 \cdot 2}$

Step 5.1.2

Multiply $- 4 \cdot 2 \cdot - 9$ .

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Step 5.1.2.1

Multiply $- 4$ by $2$ .

$c = \frac{- 6 \pm \sqrt{36 - 8 \cdot - 9}}{2 \cdot 2}$

Step 5.1.2.2

Multiply $- 8$ by $- 9$ .

$c = \frac{- 6 \pm \sqrt{36 + 72}}{2 \cdot 2}$

Step 5.1.3

Add $36$ and $72$ .

$c = \frac{- 6 \pm \sqrt{108}}{2 \cdot 2}$

Step 5.1.4

Rewrite $108$ as $6^{2} \cdot 3$ .

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Step 5.1.4.1

Factor $36$ out of $108$ .

$c = \frac{- 6 \pm \sqrt{36 (3)}}{2 \cdot 2}$

Step 5.1.4.2

Rewrite $36$ as $6^{2}$ .

$c = \frac{- 6 \pm \sqrt{6^{2} \cdot 3}}{2 \cdot 2}$

Step 5.1.5

Pull terms out from under the radical.

$c = \frac{- 6 \pm 6 \sqrt{3}}{2 \cdot 2}$

Step 5.2

Multiply $2$ by $2$ .

$c = \frac{- 6 \pm 6 \sqrt{3}}{4}$

Step 5.3

Simplify $\frac{- 6 \pm 6 \sqrt{3}}{4}$ .

$c = \frac{- 3 \pm 3 \sqrt{3}}{2}$

Step 6

The final answer is the combination of both solutions.

$c = - \frac{3 - 3 \sqrt{3}}{2}, - \frac{3 + 3 \sqrt{3}}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$c = - \frac{3 - 3 \sqrt{3}}{2}, - \frac{3 + 3 \sqrt{3}}{2}$

Decimal Form:

$c = 1.09807621 \dots, - 4.09807621 \dots$