Algebra Examples

$\sqrt{3 x^{2} + 10 x} - 5 = 0$

Step 1

Add $5$ to both sides of the equation.

$\sqrt{3 x^{2} + 10 x} = 5$

Step 2

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

${\sqrt{3 x^{2} + 10 x}}^{2} = 5^{2}$

Step 3

Simplify each side of the equation.

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 x^{2} + 10 x}$ as ${(3 x^{2} + 10 x)}^{\frac{1}{2}}$ .

${({(3 x^{2} + 10 x)}^{\frac{1}{2}})}^{2} = 5^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(3 x^{2} + 10 x)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(3 x^{2} + 10 x)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

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

${(3 x^{2} + 10 x)}^{\frac{1}{2} \cdot 2} = 5^{2}$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

${(3 x^{2} + 10 x)}^{\frac{1}{2} \cdot 2} = 5^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(3 x^{2} + 10 x)}^{1} = 5^{2}$

Step 3.2.1.2

Simplify.

$3 x^{2} + 10 x = 5^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Raise $5$ to the power of $2$ .

$3 x^{2} + 10 x = 25$

Step 4

Solve for $x$ .

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Step 4.1

Subtract $25$ from both sides of the equation.

$3 x^{2} + 10 x - 25 = 0$

Step 4.2

Factor by grouping.

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Step 4.2.1

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 $a \cdot c = 3 \cdot - 25 = - 75$ and whose sum is $b = 10$ .

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Step 4.2.1.1

Factor $10$ out of $10 x$ .

$3 x^{2} + 10 (x) - 25 = 0$

Step 4.2.1.2

Rewrite $10$ as $- 5$ plus $15$

$3 x^{2} + (- 5 + 15) x - 25 = 0$

Step 4.2.1.3

Apply the distributive property.

$3 x^{2} - 5 x + 15 x - 25 = 0$

Step 4.2.2

Factor out the greatest common factor from each group.

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Step 4.2.2.1

Group the first two terms and the last two terms.

$(3 x^{2} - 5 x) + 15 x - 25 = 0$

Step 4.2.2.2

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

$x (3 x - 5) + 5 (3 x - 5) = 0$

Step 4.2.3

Factor the polynomial by factoring out the greatest common factor, $3 x - 5$ .

$(3 x - 5) (x + 5) = 0$

Step 4.3

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

$3 x - 5 = 0$

$x + 5 = 0$

Step 4.4

Set $3 x - 5$ equal to $0$ and solve for $x$ .

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Step 4.4.1

Set $3 x - 5$ equal to $0$ .

$3 x - 5 = 0$

Step 4.4.2

Solve $3 x - 5 = 0$ for $x$ .

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Step 4.4.2.1

Add $5$ to both sides of the equation.

$3 x = 5$

Step 4.4.2.2

Divide each term in $3 x = 5$ by $3$ and simplify.

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Step 4.4.2.2.1

Divide each term in $3 x = 5$ by $3$ .

$\frac{3 x}{3} = \frac{5}{3}$

Step 4.4.2.2.2

Simplify the left side.

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Step 4.4.2.2.2.1

Cancel the common factor of $3$ .

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Step 4.4.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{5}{3}$

Step 4.4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{5}{3}$

Step 4.5

Set $x + 5$ equal to $0$ and solve for $x$ .

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Step 4.5.1

Set $x + 5$ equal to $0$ .

$x + 5 = 0$

Step 4.5.2

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 4.6

The final solution is all the values that make $(3 x - 5) (x + 5) = 0$ true.

$x = \frac{5}{3}, - 5$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5}{3}, - 5$

Decimal Form:

$x = 1.\overline{6}, - 5$

Mixed Number Form:

$x = 1 \frac{2}{3}, - 5$