Solve by Factoring square root of 15x+18=5x

$\sqrt{15 x + 18} = 5 x$

Step 1

Subtract $5 x$ from both sides of the equation.

$\sqrt{15 x + 18} - 5 x = 0$

Step 2

Factor $3$ out of $15 x + 18$ .

Factor $3$ out of $15 x$ .

$\sqrt{3 (5 x) + 18} - 5 x = 0$

Factor $3$ out of $18$ .

$\sqrt{3 (5 x) + 3 (6)} - 5 x = 0$

Factor $3$ out of $3 (5 x) + 3 (6)$ .

$\sqrt{3 (5 x + 6)} - 5 x = 0$

Step 3

Add $5 x$ to both sides of the equation.

$\sqrt{3 (5 x + 6)} = 5 x$

Step 4

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

${\sqrt{3 (5 x + 6)}}^{2} = {(5 x)}^{2}$

Step 5

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 (5 x + 6)}$ as ${(3 (5 x + 6))}^{\frac{1}{2}}$ .

${({(3 (5 x + 6))}^{\frac{1}{2}})}^{2} = {(5 x)}^{2}$

Simplify the left side.

Simplify ${({(3 (5 x + 6))}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(3 (5 x + 6))}^{\frac{1}{2}})}^{2}$ .

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

${(3 (5 x + 6))}^{\frac{1}{2} \cdot 2} = {(5 x)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(3 (5 x + 6))}^{\frac{1}{2} \cdot 2} = {(5 x)}^{2}$

Rewrite the expression.

${(3 (5 x + 6))}^{1} = {(5 x)}^{2}$

Apply the distributive property.

${(3 (5 x) + 3 \cdot 6)}^{1} = {(5 x)}^{2}$

Multiply.

Multiply $5$ by $3$ .

${(15 x + 3 \cdot 6)}^{1} = {(5 x)}^{2}$

Multiply $3$ by $6$ .

${(15 x + 18)}^{1} = {(5 x)}^{2}$

Simplify.

$15 x + 18 = {(5 x)}^{2}$

Simplify the right side.

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

Apply the product rule to $5 x$ .

$15 x + 18 = 5^{2} x^{2}$

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

$15 x + 18 = 25 x^{2}$

Step 6

Solve for $x$ .

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

$15 x + 18 - 25 x^{2} = 0$

Factor the left side of the equation.

Factor $- 1$ out of $15 x + 18 - 25 x^{2}$ .

Reorder the expression.

Move $18$ .

$15 x - 25 x^{2} + 18 = 0$

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

$- 25 x^{2} + 15 x + 18 = 0$

Factor $- 1$ out of $- 25 x^{2}$ .

$- (25 x^{2}) + 15 x + 18 = 0$

Factor $- 1$ out of $15 x$ .

$- (25 x^{2}) - (- 15 x) + 18 = 0$

Rewrite $18$ as $- 1 (- 18)$ .

$- (25 x^{2}) - (- 15 x) - 1 \cdot - 18 = 0$

Factor $- 1$ out of $- (25 x^{2}) - (- 15 x)$ .

$- (25 x^{2} - 15 x) - 1 \cdot - 18 = 0$

Factor $- 1$ out of $- (25 x^{2} - 15 x) - 1 (- 18)$ .

$- (25 x^{2} - 15 x - 18) = 0$

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 $a \cdot c = 25 \cdot - 18 = - 450$ and whose sum is $b = - 15$ .

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

$- (25 x^{2} - 15 x - 18) = 0$

Rewrite $- 15$ as $15$ plus $- 30$

$- (25 x^{2} + (15 - 30) x - 18) = 0$

Apply the distributive property.

$- (25 x^{2} + 15 x - 30 x - 18) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$- ((25 x^{2} + 15 x) - 30 x - 18) = 0$

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

$- (5 x (5 x + 3) - 6 (5 x + 3)) = 0$

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

$- ((5 x + 3) (5 x - 6)) = 0$

Remove unnecessary parentheses.

$- (5 x + 3) (5 x - 6) = 0$

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

$5 x + 3 = 0$

$5 x - 6 = 0$

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

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

$5 x + 3 = 0$

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

Subtract $3$ from both sides of the equation.

$5 x = - 3$

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

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

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

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

Set $5 x - 6$ equal to $0$ and solve for $x$ .

Set $5 x - 6$ equal to $0$ .

$5 x - 6 = 0$

Solve $5 x - 6 = 0$ for $x$ .

Add $6$ to both sides of the equation.

$5 x = 6$

Divide each term in $5 x = 6$ by $5$ and simplify.

Divide each term in $5 x = 6$ by $5$ .

$\frac{5 x}{5} = \frac{6}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x}{5} = \frac{6}{5}$

Divide $x$ by $1$ .

$x = \frac{6}{5}$

The final solution is all the values that make $- (5 x + 3) (5 x - 6) = 0$ true.

$x = - \frac{3}{5}, \frac{6}{5}$

Step 7

Exclude the solutions that do not make $\sqrt{3 (5 x + 6)} - 5 x = 0$ true.

$x = \frac{6}{5}$

Solve by Factoring square root of 15x+18=5x

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