Precalculus Examples

$4 x + 3 x^{2} - x + 2 = 38$

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

Subtract $x$ from $4 x$ .

$3 x^{2} + 3 x + 2 = 38$

Step 1.2

Subtract $38$ from both sides of the equation.

$3 x^{2} + 3 x + 2 - 38 = 0$

Step 1.3

Subtract $38$ from $2$ .

$3 x^{2} + 3 x - 36 = 0$

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 $a = 3$ , $b = 3$ , and $c = - 36$ into the quadratic formula and solve for $x$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (3 \cdot - 36)}}{2 \cdot 3}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $3$ to the power of $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 3 \cdot - 36}}{2 \cdot 3}$

Step 4.1.2

Multiply $- 4 \cdot 3 \cdot - 36$ .

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Step 4.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{- 3 \pm \sqrt{9 - 12 \cdot - 36}}{2 \cdot 3}$

Step 4.1.2.2

Multiply $- 12$ by $- 36$ .

$x = \frac{- 3 \pm \sqrt{9 + 432}}{2 \cdot 3}$

Step 4.1.3

Add $9$ and $432$ .

$x = \frac{- 3 \pm \sqrt{441}}{2 \cdot 3}$

Step 4.1.4

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

$x = \frac{- 3 \pm \sqrt{21^{2}}}{2 \cdot 3}$

Step 4.1.5

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

$x = \frac{- 3 \pm 21}{2 \cdot 3}$

Step 4.2

Multiply $2$ by $3$ .

$x = \frac{- 3 \pm 21}{6}$

Step 4.3

Simplify $\frac{- 3 \pm 21}{6}$ .

$x = \frac{- 1 \pm 7}{2}$

Step 5

The final answer is the combination of both solutions.

$x = 3, - 4$