Solve for x square root of 15-x- square root of 6x=-3

$\sqrt{15 - x} - \sqrt{6 x} = - 3$

Add $\sqrt{6 x}$ to both sides of the equation.

$\sqrt{15 - x} = - 3 + \sqrt{6 x}$

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

${\sqrt{15 - x}}^{2} = {(- 3 + \sqrt{6 x})}^{2}$

Simplify each side of the equation.

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

${({(15 - x)}^{\frac{1}{2}})}^{2} = {(- 3 + \sqrt{6 x})}^{2}$

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

${(15 - x)}^{1} = {(- 3 + \sqrt{6 x})}^{2}$

Simplify.

$15 - x = {(- 3 + \sqrt{6 x})}^{2}$

Simplify the right side.

Simplify ${(- 3 + \sqrt{6 x})}^{2}$ .

Rewrite ${(- 3 + \sqrt{6 x})}^{2}$ as $(- 3 + \sqrt{6 x}) (- 3 + \sqrt{6 x})$ .

$15 - x = (- 3 + \sqrt{6 x}) (- 3 + \sqrt{6 x})$

Expand $(- 3 + \sqrt{6 x}) (- 3 + \sqrt{6 x})$ using the FOIL Method.

Apply the distributive property.

$15 - x = - 3 (- 3 + \sqrt{6 x}) + \sqrt{6 x} (- 3 + \sqrt{6 x})$

Apply the distributive property.

$15 - x = - 3 \cdot - 3 - 3 \sqrt{6 x} + \sqrt{6 x} (- 3 + \sqrt{6 x})$

Apply the distributive property.

$15 - x = - 3 \cdot - 3 - 3 \sqrt{6 x} + \sqrt{6 x} \cdot - 3 + \sqrt{6 x} \sqrt{6 x}$

Simplify and combine like terms.

Simplify each term.

Multiply $- 3$ by $- 3$ .

$15 - x = 9 - 3 \sqrt{6 x} + \sqrt{6 x} \cdot - 3 + \sqrt{6 x} \sqrt{6 x}$

Move $- 3$ to the left of $\sqrt{6 x}$ .

$15 - x = 9 - 3 \sqrt{6 x} - 3 \cdot \sqrt{6 x} + \sqrt{6 x} \sqrt{6 x}$

Multiply $\sqrt{6 x} \sqrt{6 x}$ .

Raise $\sqrt{6 x}$ to the power of $1$ .

$15 - x = 9 - 3 \sqrt{6 x} - 3 \sqrt{6 x} + {\sqrt{6 x}}^{1} \sqrt{6 x}$

Raise $\sqrt{6 x}$ to the power of $1$ .

$15 - x = 9 - 3 \sqrt{6 x} - 3 \sqrt{6 x} + {\sqrt{6 x}}^{1} {\sqrt{6 x}}^{1}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$15 - x = 9 - 3 \sqrt{6 x} - 3 \sqrt{6 x} + {\sqrt{6 x}}^{1 + 1}$

Add $1$ and $1$ .

$15 - x = 9 - 3 \sqrt{6 x} - 3 \sqrt{6 x} + {\sqrt{6 x}}^{2}$

Rewrite ${\sqrt{6 x}}^{2}$ as $6 x$ .

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

$15 - x = 9 - 3 \sqrt{6 x} - 3 \sqrt{6 x} + {({(6 x)}^{\frac{1}{2}})}^{2}$

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

$15 - x = 9 - 3 \sqrt{6 x} - 3 \sqrt{6 x} + {(6 x)}^{\frac{1}{2} \cdot 2}$

Combine $\frac{1}{2}$ and $2$ .

$15 - x = 9 - 3 \sqrt{6 x} - 3 \sqrt{6 x} + {(6 x)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$15 - x = 9 - 3 \sqrt{6 x} - 3 \sqrt{6 x} + {(6 x)}^{\frac{2}{2}}$

Rewrite the expression.

$15 - x = 9 - 3 \sqrt{6 x} - 3 \sqrt{6 x} + {(6 x)}^{1}$

Simplify.

$15 - x = 9 - 3 \sqrt{6 x} - 3 \sqrt{6 x} + 6 x$

Subtract $3 \sqrt{6 x}$ from $- 3 \sqrt{6 x}$ .

$15 - x = 9 - 6 \sqrt{6 x} + 6 x$

Solve for $- 6 \sqrt{6 x}$ .

Rewrite the equation as $9 - 6 \sqrt{6 x} + 6 x = 15 - x$ .

$9 - 6 \sqrt{6 x} + 6 x = 15 - x$

Move all terms not containing $- 6 \sqrt{6 x}$ to the right side of the equation.

Subtract $9$ from both sides of the equation.

$- 6 \sqrt{6 x} + 6 x = 15 - x - 9$

Subtract $6 x$ from both sides of the equation.

$- 6 \sqrt{6 x} = 15 - x - 9 - 6 x$

Subtract $9$ from $15$ .

$- 6 \sqrt{6 x} = - x + 6 - 6 x$

Subtract $6 x$ from $- x$ .

$- 6 \sqrt{6 x} = - 7 x + 6$

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

${(- 6 \sqrt{6 x})}^{2} = {(- 7 x + 6)}^{2}$

Simplify each side of the equation.

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

${(- 6 {(6 x)}^{\frac{1}{2}})}^{2} = {(- 7 x + 6)}^{2}$

Simplify the left side.

Simplify ${(- 6 {(6 x)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $6 x$ .

${(- 6 (6^{\frac{1}{2}} x^{\frac{1}{2}}))}^{2} = {(- 7 x + 6)}^{2}$

Multiply $- 6 \cdot 6^{\frac{1}{2}}$ .

Factor out negative.

${(- (6 \cdot 6^{\frac{1}{2}}) x^{\frac{1}{2}})}^{2} = {(- 7 x + 6)}^{2}$

Raise $6$ to the power of $1$ .

${(- (6^{1} \cdot 6^{\frac{1}{2}}) x^{\frac{1}{2}})}^{2} = {(- 7 x + 6)}^{2}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

${(- 6^{1 + \frac{1}{2}} x^{\frac{1}{2}})}^{2} = {(- 7 x + 6)}^{2}$

Write $1$ as a fraction with a common denominator.

${(- 6^{\frac{2}{2} + \frac{1}{2}} x^{\frac{1}{2}})}^{2} = {(- 7 x + 6)}^{2}$

Combine the numerators over the common denominator.

${(- 6^{\frac{2 + 1}{2}} x^{\frac{1}{2}})}^{2} = {(- 7 x + 6)}^{2}$

Add $2$ and $1$ .

${(- 6^{\frac{3}{2}} x^{\frac{1}{2}})}^{2} = {(- 7 x + 6)}^{2}$

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $- 6^{\frac{3}{2}} x^{\frac{1}{2}}$ .

${(- 6^{\frac{3}{2}})}^{2} {(x^{\frac{1}{2}})}^{2} = {(- 7 x + 6)}^{2}$

Apply the product rule to $- 6^{\frac{3}{2}}$ .

${(- 1)}^{2} {(6^{\frac{3}{2}})}^{2} {(x^{\frac{1}{2}})}^{2} = {(- 7 x + 6)}^{2}$

Raise $- 1$ to the power of $2$ .

$1 {(6^{\frac{3}{2}})}^{2} {(x^{\frac{1}{2}})}^{2} = {(- 7 x + 6)}^{2}$

Multiply ${(6^{\frac{3}{2}})}^{2}$ by $1$ .

${(6^{\frac{3}{2}})}^{2} {(x^{\frac{1}{2}})}^{2} = {(- 7 x + 6)}^{2}$

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

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

$6^{\frac{3}{2} \cdot 2} {(x^{\frac{1}{2}})}^{2} = {(- 7 x + 6)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$6^{\frac{3}{2} \cdot 2} {(x^{\frac{1}{2}})}^{2} = {(- 7 x + 6)}^{2}$

Rewrite the expression.

$6^{3} {(x^{\frac{1}{2}})}^{2} = {(- 7 x + 6)}^{2}$

Raise $6$ to the power of $3$ .

$216 {(x^{\frac{1}{2}})}^{2} = {(- 7 x + 6)}^{2}$

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

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

$216 x^{\frac{1}{2} \cdot 2} = {(- 7 x + 6)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$216 x^{\frac{1}{2} \cdot 2} = {(- 7 x + 6)}^{2}$

Rewrite the expression.

$216 x^{1} = {(- 7 x + 6)}^{2}$

Simplify.

$216 x = {(- 7 x + 6)}^{2}$

Simplify the right side.

Simplify ${(- 7 x + 6)}^{2}$ .

Rewrite ${(- 7 x + 6)}^{2}$ as $(- 7 x + 6) (- 7 x + 6)$ .

$216 x = (- 7 x + 6) (- 7 x + 6)$

Expand $(- 7 x + 6) (- 7 x + 6)$ using the FOIL Method.

Apply the distributive property.

$216 x = - 7 x (- 7 x + 6) + 6 (- 7 x + 6)$

Apply the distributive property.

$216 x = - 7 x (- 7 x) - 7 x \cdot 6 + 6 (- 7 x + 6)$

Apply the distributive property.

$216 x = - 7 x (- 7 x) - 7 x \cdot 6 + 6 (- 7 x) + 6 \cdot 6$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$216 x = - 7 \cdot - 7 x \cdot x - 7 x \cdot 6 + 6 (- 7 x) + 6 \cdot 6$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$216 x = - 7 \cdot - 7 (x \cdot x) - 7 x \cdot 6 + 6 (- 7 x) + 6 \cdot 6$

Multiply $x$ by $x$ .

$216 x = - 7 \cdot - 7 x^{2} - 7 x \cdot 6 + 6 (- 7 x) + 6 \cdot 6$

Multiply $- 7$ by $- 7$ .

$216 x = 49 x^{2} - 7 x \cdot 6 + 6 (- 7 x) + 6 \cdot 6$

Multiply $6$ by $- 7$ .

$216 x = 49 x^{2} - 42 x + 6 (- 7 x) + 6 \cdot 6$

Multiply $- 7$ by $6$ .

$216 x = 49 x^{2} - 42 x - 42 x + 6 \cdot 6$

Multiply $6$ by $6$ .

$216 x = 49 x^{2} - 42 x - 42 x + 36$

Subtract $42 x$ from $- 42 x$ .

$216 x = 49 x^{2} - 84 x + 36$

Solve for $x$ .

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

$49 x^{2} - 84 x + 36 = 216 x$

Move all terms containing $x$ to the left side of the equation.

Subtract $216 x$ from both sides of the equation.

$49 x^{2} - 84 x + 36 - 216 x = 0$

Subtract $216 x$ from $- 84 x$ .

$49 x^{2} - 300 x + 36 = 0$

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 $a \cdot c = 49 \cdot 36 = 1764$ and whose sum is $b = - 300$ .

Factor $- 300$ out of $- 300 x$ .

$49 x^{2} - 300 x + 36 = 0$

Rewrite $- 300$ as $- 6$ plus $- 294$

$49 x^{2} + (- 6 - 294) x + 36 = 0$

Apply the distributive property.

$49 x^{2} - 6 x - 294 x + 36 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(49 x^{2} - 6 x) - 294 x + 36 = 0$

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

$x (49 x - 6) - 6 (49 x - 6) = 0$

Factor the polynomial by factoring out the greatest common factor, $49 x - 6$ .

$(49 x - 6) (x - 6) = 0$

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

$49 x - 6 = 0$

$x - 6 = 0$

Set $49 x - 6$ equal to $0$ and solve for $x$ .

Set $49 x - 6$ equal to $0$ .

$49 x - 6 = 0$

Solve $49 x - 6 = 0$ for $x$ .

Add $6$ to both sides of the equation.

$49 x = 6$

Divide each term in $49 x = 6$ by $49$ and simplify.

Divide each term in $49 x = 6$ by $49$ .

$\frac{49 x}{49} = \frac{6}{49}$

Simplify the left side.

Cancel the common factor of $49$ .

Cancel the common factor.

$\frac{49 x}{49} = \frac{6}{49}$

Divide $x$ by $1$ .

$x = \frac{6}{49}$

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

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

$x - 6 = 0$

Add $6$ to both sides of the equation.

$x = 6$

The final solution is all the values that make $(49 x - 6) (x - 6) = 0$ true.

$x = \frac{6}{49}, 6$

Exclude the solutions that do not make $\sqrt{15 - x} - \sqrt{6 x} = - 3$ true.

$x = 6$

Solve for x square root of 15-x- square root of 6x=-3

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