Precalculus Examples

$2 x^{4} + 14 x^{3} - 35 x^{2}$

Step 1

Set $2 x^{4} + 14 x^{3} - 35 x^{2}$ equal to $0$ .

$2 x^{4} + 14 x^{3} - 35 x^{2} = 0$

Step 2

Solve for $x$ .

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Step 2.1

Factor $x^{2}$ out of $2 x^{4} + 14 x^{3} - 35 x^{2}$ .

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Step 2.1.1

Factor $x^{2}$ out of $2 x^{4}$ .

$x^{2} (2 x^{2}) + 14 x^{3} - 35 x^{2} = 0$

Step 2.1.2

Factor $x^{2}$ out of $14 x^{3}$ .

$x^{2} (2 x^{2}) + x^{2} (14 x) - 35 x^{2} = 0$

Step 2.1.3

Factor $x^{2}$ out of $- 35 x^{2}$ .

$x^{2} (2 x^{2}) + x^{2} (14 x) + x^{2} \cdot - 35 = 0$

Step 2.1.4

Factor $x^{2}$ out of $x^{2} (2 x^{2}) + x^{2} (14 x)$ .

$x^{2} (2 x^{2} + 14 x) + x^{2} \cdot - 35 = 0$

Step 2.1.5

Factor $x^{2}$ out of $x^{2} (2 x^{2} + 14 x) + x^{2} \cdot - 35$ .

$x^{2} (2 x^{2} + 14 x - 35) = 0$

Step 2.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x^{2} = 0$

$2 x^{2} + 14 x - 35 = 0$

Step 2.3

Set $x^{2}$ equal to $0$ and solve for $x$ .

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Step 2.3.1

Set $x^{2}$ equal to $0$ .

$x^{2} = 0$

Step 2.3.2

Solve $x^{2} = 0$ for $x$ .

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Step 2.3.2.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{0}$

Step 2.3.2.2

Simplify $\pm \sqrt{0}$ .

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Step 2.3.2.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 2.3.2.2.2

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

$x = \pm 0$

Step 2.3.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 2.4

Set $2 x^{2} + 14 x - 35$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $2 x^{2} + 14 x - 35$ equal to $0$ .

$2 x^{2} + 14 x - 35 = 0$

Step 2.4.2

Solve $2 x^{2} + 14 x - 35 = 0$ for $x$ .

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Step 2.4.2.1

Use the quadratic formula to find the solutions.

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

Step 2.4.2.2

Substitute the values $a = 2$ , $b = 14$ , and $c = - 35$ into the quadratic formula and solve for $x$ .

$\frac{- 14 \pm \sqrt{14^{2} - 4 \cdot (2 \cdot - 35)}}{2 \cdot 2}$

Step 2.4.2.3

Simplify.

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Step 2.4.2.3.1

Simplify the numerator.

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Step 2.4.2.3.1.1

Raise $14$ to the power of $2$ .

$x = \frac{- 14 \pm \sqrt{196 - 4 \cdot 2 \cdot - 35}}{2 \cdot 2}$

Step 2.4.2.3.1.2

Multiply $- 4 \cdot 2 \cdot - 35$ .

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Step 2.4.2.3.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{- 14 \pm \sqrt{196 - 8 \cdot - 35}}{2 \cdot 2}$

Step 2.4.2.3.1.2.2

Multiply $- 8$ by $- 35$ .

$x = \frac{- 14 \pm \sqrt{196 + 280}}{2 \cdot 2}$

Step 2.4.2.3.1.3

Add $196$ and $280$ .

$x = \frac{- 14 \pm \sqrt{476}}{2 \cdot 2}$

Step 2.4.2.3.1.4

Rewrite $476$ as $2^{2} \cdot 119$ .

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Step 2.4.2.3.1.4.1

Factor $4$ out of $476$ .

$x = \frac{- 14 \pm \sqrt{4 (119)}}{2 \cdot 2}$

Step 2.4.2.3.1.4.2

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

$x = \frac{- 14 \pm \sqrt{2^{2} \cdot 119}}{2 \cdot 2}$

Step 2.4.2.3.1.5

Pull terms out from under the radical.

$x = \frac{- 14 \pm 2 \sqrt{119}}{2 \cdot 2}$

Step 2.4.2.3.2

Multiply $2$ by $2$ .

$x = \frac{- 14 \pm 2 \sqrt{119}}{4}$

Step 2.4.2.3.3

Simplify $\frac{- 14 \pm 2 \sqrt{119}}{4}$ .

$x = \frac{- 7 \pm \sqrt{119}}{2}$

Step 2.4.2.4

The final answer is the combination of both solutions.

$x = - \frac{7 - \sqrt{119}}{2}, - \frac{7 + \sqrt{119}}{2}$

Step 2.5

The final solution is all the values that make $x^{2} (2 x^{2} + 14 x - 35) = 0$ true.

$x = 0, - \frac{7 - \sqrt{119}}{2}, - \frac{7 + \sqrt{119}}{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = 0, - \frac{7 - \sqrt{119}}{2}, - \frac{7 + \sqrt{119}}{2}$

Decimal Form:

$x = 0, 1.95435605 \dots, - 8.95435605 \dots$

Step 4