Precalculus Examples

$x (3 x - 2) = 4$

Step 1

Move all terms to the left side of the equation and simplify.

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Step 1.1

Simplify the left side.

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Step 1.1.1

Simplify $x (3 x - 2)$ .

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Step 1.1.1.1

Simplify by multiplying through.

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Step 1.1.1.1.1

Apply the distributive property.

$x (3 x) + x \cdot - 2 = 4$

Step 1.1.1.1.2

Reorder.

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Step 1.1.1.1.2.1

Rewrite using the commutative property of multiplication.

$3 x \cdot x + x \cdot - 2 = 4$

Step 1.1.1.1.2.2

Move $- 2$ to the left of $x$ .

$3 x \cdot x - 2 \cdot x = 4$

Step 1.1.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 1.1.1.2.1

Move $x$ .

$3 (x \cdot x) - 2 \cdot x = 4$

Step 1.1.1.2.2

Multiply $x$ by $x$ .

$3 x^{2} - 2 \cdot x = 4$

$3 x^{2} - 2 x = 4$

Step 1.2

Subtract $4$ from both sides of the equation.

$3 x^{2} - 2 x - 4 = 0$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values $a = 3$ , $b = - 2$ , and $c = - 4$ into the quadratic formula and solve for $x$ .

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

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 3 \cdot - 4}}{2 \cdot 3}$

Step 4.1.2

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

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Step 4.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{2 \pm \sqrt{4 - 12 \cdot - 4}}{2 \cdot 3}$

Step 4.1.2.2

Multiply $- 12$ by $- 4$ .

$x = \frac{2 \pm \sqrt{4 + 48}}{2 \cdot 3}$

Step 4.1.3

Add $4$ and $48$ .

$x = \frac{2 \pm \sqrt{52}}{2 \cdot 3}$

Step 4.1.4

Rewrite $52$ as $2^{2} \cdot 13$ .

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Step 4.1.4.1

Factor $4$ out of $52$ .

$x = \frac{2 \pm \sqrt{4 (13)}}{2 \cdot 3}$

Step 4.1.4.2

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

$x = \frac{2 \pm \sqrt{2^{2} \cdot 13}}{2 \cdot 3}$

Step 4.1.5

Pull terms out from under the radical.

$x = \frac{2 \pm 2 \sqrt{13}}{2 \cdot 3}$

Step 4.2

Multiply $2$ by $3$ .

$x = \frac{2 \pm 2 \sqrt{13}}{6}$

Step 4.3

Simplify $\frac{2 \pm 2 \sqrt{13}}{6}$ .

$x = \frac{1 \pm \sqrt{13}}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1 \pm \sqrt{13}}{3}$

Decimal Form:

$x = 1.53518375 \dots, - 0.86851709 \dots$