Algebra Examples

$5 x = \sqrt{10 + 15 x}$

Step 1

Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.

$\sqrt{10 + 15 x} = 5 x$

Step 2

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

${\sqrt{10 + 15 x}}^{2} = {(5 x)}^{2}$

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 $\sqrt{10 + 15 x}$ as ${(10 + 15 x)}^{\frac{1}{2}}$ .

${({(10 + 15 x)}^{\frac{1}{2}})}^{2} = {(5 x)}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(10 + 15 x)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(10 + 15 x)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

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

${(10 + 15 x)}^{\frac{1}{2} \cdot 2} = {(5 x)}^{2}$

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.

${(10 + 15 x)}^{\frac{1}{2} \cdot 2} = {(5 x)}^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(10 + 15 x)}^{1} = {(5 x)}^{2}$

Step 3.2.1.2

Simplify.

$10 + 15 x = {(5 x)}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${(5 x)}^{2}$ .

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Step 3.3.1.1

Apply the product rule to $5 x$ .

$10 + 15 x = 5^{2} x^{2}$

Step 3.3.1.2

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

$10 + 15 x = 25 x^{2}$

Step 4

Solve for $x$ .

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Step 4.1

Subtract $25 x^{2}$ from both sides of the equation.

$10 + 15 x - 25 x^{2} = 0$

Step 4.2

Factor the left side of the equation.

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Step 4.2.1

Factor $- 5$ out of $10 + 15 x - 25 x^{2}$ .

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Step 4.2.1.1

Reorder the expression.

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Step 4.2.1.1.1

Move $10$ .

$15 x - 25 x^{2} + 10 = 0$

Step 4.2.1.1.2

Reorder $15 x$ and $- 25 x^{2}$ .

$- 25 x^{2} + 15 x + 10 = 0$

Step 4.2.1.2

Factor $- 5$ out of $- 25 x^{2}$ .

$- 5 (5 x^{2}) + 15 x + 10 = 0$

Step 4.2.1.3

Factor $- 5$ out of $15 x$ .

$- 5 (5 x^{2}) - 5 (- 3 x) + 10 = 0$

Step 4.2.1.4

Factor $- 5$ out of $10$ .

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

Step 4.2.1.5

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

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

Step 4.2.1.6

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

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

Step 4.2.2

Factor.

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Step 4.2.2.1

Factor by grouping.

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Step 4.2.2.1.1

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 $a \cdot c = 5 \cdot - 2 = - 10$ and whose sum is $b = - 3$ .

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Step 4.2.2.1.1.1

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

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

Step 4.2.2.1.1.2

Rewrite $- 3$ as $2$ plus $- 5$

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

Step 4.2.2.1.1.3

Apply the distributive property.

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

Step 4.2.2.1.2

Factor out the greatest common factor from each group.

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Step 4.2.2.1.2.1

Group the first two terms and the last two terms.

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

Step 4.2.2.1.2.2

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

$- 5 (x (5 x + 2) - (5 x + 2)) = 0$

Step 4.2.2.1.3

Factor the polynomial by factoring out the greatest common factor, $5 x + 2$ .

$- 5 ((5 x + 2) (x - 1)) = 0$

Step 4.2.2.2

Remove unnecessary parentheses.

$- 5 (5 x + 2) (x - 1) = 0$

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 $0$ .

$5 x + 2 = 0$

$x - 1 = 0$

Step 4.4

Set $5 x + 2$ equal to $0$ and solve for $x$ .

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Step 4.4.1

Set $5 x + 2$ equal to $0$ .

$5 x + 2 = 0$

Step 4.4.2

Solve $5 x + 2 = 0$ for $x$ .

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Step 4.4.2.1

Subtract $2$ from both sides of the equation.

$5 x = - 2$

Step 4.4.2.2

Divide each term in $5 x = - 2$ by $5$ and simplify.

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Step 4.4.2.2.1

Divide each term in $5 x = - 2$ by $5$ .

$\frac{5 x}{5} = \frac{- 2}{5}$

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 $5$ .

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Step 4.4.2.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{- 2}{5}$

Step 4.4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 2}{5}$

Step 4.4.2.2.3

Simplify the right side.

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Step 4.4.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{2}{5}$

Step 4.5

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 4.5.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 4.5.2

Add $1$ to both sides of the equation.

$x = 1$

Step 4.6

The final solution is all the values that make $- 5 (5 x + 2) (x - 1) = 0$ true.

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

Step 5

Exclude the solutions that do not make $5 x = \sqrt{10 + 15 x}$ true.

$x = 1$