Algebra Examples

$f (x) = \sqrt{2 x^{3} - 5 x^{2} - 3 x}$

Step 1

Set $\sqrt{2 x^{3} - 5 x^{2} - 3 x}$ equal to $0$ .

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

Step 2

Solve for $x$ .

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Step 2.1

To remove the radical on the left side of the equation, square both sides of the equation.

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

Step 2.2

Simplify each side of the equation.

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Step 2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 x^{3} - 5 x^{2} - 3 x}$ as ${(2 x^{3} - 5 x^{2} - 3 x)}^{\frac{1}{2}}$ .

${({(2 x^{3} - 5 x^{2} - 3 x)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Simplify ${({(2 x^{3} - 5 x^{2} - 3 x)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.2.1.1

Multiply the exponents in ${({(2 x^{3} - 5 x^{2} - 3 x)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(2 x^{3} - 5 x^{2} - 3 x)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 2.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.2.1.1.2.1

Cancel the common factor.

${(2 x^{3} - 5 x^{2} - 3 x)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 2.2.2.1.1.2.2

Rewrite the expression.

${(2 x^{3} - 5 x^{2} - 3 x)}^{1} = 0^{2}$

Step 2.2.2.1.2

Simplify.

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

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Raising $0$ to any positive power yields $0$ .

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

Step 2.3

Solve for $x$ .

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Step 2.3.1

Factor the left side of the equation.

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Step 2.3.1.1

Factor $x$ out of $2 x^{3} - 5 x^{2} - 3 x$ .

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Step 2.3.1.1.1

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

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

Step 2.3.1.1.2

Factor $x$ out of $- 5 x^{2}$ .

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

Step 2.3.1.1.3

Factor $x$ out of $- 3 x$ .

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

Step 2.3.1.1.4

Factor $x$ out of $x (2 x^{2}) + x (- 5 x)$ .

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

Step 2.3.1.1.5

Factor $x$ out of $x (2 x^{2} - 5 x) + x \cdot - 3$ .

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

Step 2.3.1.2

Factor.

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Step 2.3.1.2.1

Factor by grouping.

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Step 2.3.1.2.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 3 = - 6$ and whose sum is $b = - 5$ .

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Step 2.3.1.2.1.1.1

Factor $- 5$ out of $- 5 x$ .

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

Step 2.3.1.2.1.1.2

Rewrite $- 5$ as $1$ plus $- 6$

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

Step 2.3.1.2.1.1.3

Apply the distributive property.

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

Step 2.3.1.2.1.1.4

Multiply $x$ by $1$ .

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

Step 2.3.1.2.1.2

Factor out the greatest common factor from each group.

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Step 2.3.1.2.1.2.1

Group the first two terms and the last two terms.

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

Step 2.3.1.2.1.2.2

Factor out the greatest common factor (GCF) from each group.

$x (x (2 x + 1) - 3 (2 x + 1)) = 0$

Step 2.3.1.2.1.3

Factor the polynomial by factoring out the greatest common factor, $2 x + 1$ .

$x ((2 x + 1) (x - 3)) = 0$

Step 2.3.1.2.2

Remove unnecessary parentheses.

$x (2 x + 1) (x - 3) = 0$

Step 2.3.2

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

$x = 0$

$2 x + 1 = 0$

$x - 3 = 0$

Step 2.3.3

Set $x$ equal to $0$ .

$x = 0$

Step 2.3.4

Set $2 x + 1$ equal to $0$ and solve for $x$ .

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Step 2.3.4.1

Set $2 x + 1$ equal to $0$ .

$2 x + 1 = 0$

Step 2.3.4.2

Solve $2 x + 1 = 0$ for $x$ .

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Step 2.3.4.2.1

Subtract $1$ from both sides of the equation.

$2 x = - 1$

Step 2.3.4.2.2

Divide each term in $2 x = - 1$ by $2$ and simplify.

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Step 2.3.4.2.2.1

Divide each term in $2 x = - 1$ by $2$ .

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

Step 2.3.4.2.2.2

Simplify the left side.

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Step 2.3.4.2.2.2.1

Cancel the common factor of $2$ .

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Step 2.3.4.2.2.2.1.1

Cancel the common factor.

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

Step 2.3.4.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 2.3.4.2.2.3

Simplify the right side.

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Step 2.3.4.2.2.3.1

Move the negative in front of the fraction.

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

Step 2.3.5

Set $x - 3$ equal to $0$ and solve for $x$ .

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Step 2.3.5.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 2.3.5.2

Add $3$ to both sides of the equation.

$x = 3$

Step 2.3.6

The final solution is all the values that make $x (2 x + 1) (x - 3) = 0$ true.

$x = 0, - \frac{1}{2}, 3$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = 0, - \frac{1}{2}, 3$

Decimal Form:

$x = 0, - 0.5, 3$

Step 4