Finite Math Examples

$(2 x^{2} - 6 - 3 x + 1) (3)$

Step 1

Set $(2 x^{2} - 6 - 3 x + 1) (3)$ equal to $0$ .

$(2 x^{2} - 6 - 3 x + 1) (3) = 0$

Step 2

Simplify the left side.

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Step 2.1

Simplify $(2 x^{2} - 6 - 3 x + 1) (3)$ .

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Step 2.1.1

Add $- 6$ and $1$ .

$(2 x^{2} - 3 x - 5) \cdot 3 = 0$

Step 2.1.2

Apply the distributive property.

$2 x^{2} \cdot 3 - 3 x \cdot 3 - 5 \cdot 3 = 0$

Step 2.1.3

Simplify.

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Step 2.1.3.1

Multiply $3$ by $2$ .

$6 x^{2} - 3 x \cdot 3 - 5 \cdot 3 = 0$

Step 2.1.3.2

Multiply $3$ by $- 3$ .

$6 x^{2} - 9 x - 5 \cdot 3 = 0$

Step 2.1.3.3

Multiply $- 5$ by $3$ .

$6 x^{2} - 9 x - 15 = 0$

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 $a = 6$ , $b = - 9$ , and $c = - 15$ into the quadratic formula and solve for $x$ .

$\frac{9 \pm \sqrt{{(- 9)}^{2} - 4 \cdot (6 \cdot - 15)}}{2 \cdot 6}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $- 9$ to the power of $2$ .

$x = \frac{9 \pm \sqrt{81 - 4 \cdot 6 \cdot - 15}}{2 \cdot 6}$

Step 5.1.2

Multiply $- 4 \cdot 6 \cdot - 15$ .

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Step 5.1.2.1

Multiply $- 4$ by $6$ .

$x = \frac{9 \pm \sqrt{81 - 24 \cdot - 15}}{2 \cdot 6}$

Step 5.1.2.2

Multiply $- 24$ by $- 15$ .

$x = \frac{9 \pm \sqrt{81 + 360}}{2 \cdot 6}$

Step 5.1.3

Add $81$ and $360$ .

$x = \frac{9 \pm \sqrt{441}}{2 \cdot 6}$

Step 5.1.4

Rewrite $441$ as $21^{2}$ .

$x = \frac{9 \pm \sqrt{21^{2}}}{2 \cdot 6}$

Step 5.1.5

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{9 \pm 21}{2 \cdot 6}$

Step 5.2

Multiply $2$ by $6$ .

$x = \frac{9 \pm 21}{12}$

Step 5.3

Simplify $\frac{9 \pm 21}{12}$ .

$x = \frac{3 \pm 7}{4}$

Step 6

The final answer is the combination of both solutions.

$x = \frac{5}{2}, - 1$