Find All Complex Solutions 5x = square root of 10+15x

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

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

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

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

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

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10 + 15 x}$ as ${(10 + 15 x)}^{\frac{1}{2}}$ .

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

Apply the product rule to $5 x$ .

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

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

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

Solve for $x$ .

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

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

Factor the left side of the equation.

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

Reorder the expression.

Move $10$ .

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

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

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

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

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

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

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

Factor $- 5$ out of $10$ .

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

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

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

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

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

Factor.

Factor by grouping.

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 $b = - 3$ .

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

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

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

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

Apply the distributive property.

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

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

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

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

Remove unnecessary parentheses.

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

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

$5 x + 2 = 0$

$x - 1 = 0$

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

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

$5 x + 2 = 0$

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

Subtract $2$ from both sides of the equation.

$5 x = - 2$

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

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

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

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

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

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

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

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

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

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

$x = 1$

Find All Complex Solutions 5x = square root of 10+15x

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