Solve the Rational Equation for x square root of 1-5x=1+ square root of 6-x

$\sqrt{1 - 5 x} = 1 + \sqrt{6 - x}$

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

${\sqrt{1 - 5 x}}^{2} = {(1 + \sqrt{6 - x})}^{2}$

Simplify each side of the equation.

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

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

Simplify the left side.

Simplify ${({(1 - 5 x)}^{\frac{1}{2}})}^{2}$ .

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

${(1 - 5 x)}^{1} = {(1 + \sqrt{6 - x})}^{2}$

Simplify.

$1 - 5 x = {(1 + \sqrt{6 - x})}^{2}$

Simplify the right side.

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

Rewrite ${(1 + \sqrt{6 - x})}^{2}$ as $(1 + \sqrt{6 - x}) (1 + \sqrt{6 - x})$ .

$1 - 5 x = (1 + \sqrt{6 - x}) (1 + \sqrt{6 - x})$

Expand $(1 + \sqrt{6 - x}) (1 + \sqrt{6 - x})$ using the FOIL Method.

Apply the distributive property.

$1 - 5 x = 1 (1 + \sqrt{6 - x}) + \sqrt{6 - x} (1 + \sqrt{6 - x})$

Apply the distributive property.

$1 - 5 x = 1 \cdot 1 + 1 \sqrt{6 - x} + \sqrt{6 - x} (1 + \sqrt{6 - x})$

Apply the distributive property.

$1 - 5 x = 1 \cdot 1 + 1 \sqrt{6 - x} + \sqrt{6 - x} \cdot 1 + \sqrt{6 - x} \sqrt{6 - x}$

Simplify and combine like terms.

Simplify each term.

Multiply $1$ by $1$ .

$1 - 5 x = 1 + 1 \sqrt{6 - x} + \sqrt{6 - x} \cdot 1 + \sqrt{6 - x} \sqrt{6 - x}$

Multiply $\sqrt{6 - x}$ by $1$ .

$1 - 5 x = 1 + \sqrt{6 - x} + \sqrt{6 - x} \cdot 1 + \sqrt{6 - x} \sqrt{6 - x}$

Multiply $\sqrt{6 - x}$ by $1$ .

$1 - 5 x = 1 + \sqrt{6 - x} + \sqrt{6 - x} + \sqrt{6 - x} \sqrt{6 - x}$

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

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

$1 - 5 x = 1 + \sqrt{6 - x} + \sqrt{6 - x} + {\sqrt{6 - x}}^{1} \sqrt{6 - x}$

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

$1 - 5 x = 1 + \sqrt{6 - x} + \sqrt{6 - x} + {\sqrt{6 - x}}^{1} {\sqrt{6 - x}}^{1}$

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

$1 - 5 x = 1 + \sqrt{6 - x} + \sqrt{6 - x} + {\sqrt{6 - x}}^{1 + 1}$

Add $1$ and $1$ .

$1 - 5 x = 1 + \sqrt{6 - x} + \sqrt{6 - x} + {\sqrt{6 - x}}^{2}$

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$1 - 5 x = 1 + \sqrt{6 - x} + \sqrt{6 - x} + {(6 - x)}^{1}$

Simplify.

$1 - 5 x = 1 + \sqrt{6 - x} + \sqrt{6 - x} + 6 - x$

Add $1$ and $6$ .

$1 - 5 x = 7 + \sqrt{6 - x} + \sqrt{6 - x} - x$

Add $\sqrt{6 - x}$ and $\sqrt{6 - x}$ .

$1 - 5 x = 7 + 2 \sqrt{6 - x} - x$

Solve for $2 \sqrt{6 - x}$ .

Rewrite the equation as $7 + 2 \sqrt{6 - x} - x = 1 - 5 x$ .

$7 + 2 \sqrt{6 - x} - x = 1 - 5 x$

Move all terms not containing $2 \sqrt{6 - x}$ to the right side of the equation.

Subtract $7$ from both sides of the equation.

$2 \sqrt{6 - x} - x = 1 - 5 x - 7$

Add $x$ to both sides of the equation.

$2 \sqrt{6 - x} = 1 - 5 x - 7 + x$

Subtract $7$ from $1$ .

$2 \sqrt{6 - x} = - 5 x - 6 + x$

Add $- 5 x$ and $x$ .

$2 \sqrt{6 - x} = - 4 x - 6$

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

${(2 \sqrt{6 - x})}^{2} = {(- 4 x - 6)}^{2}$

Simplify each side of the equation.

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

${(2 {(6 - x)}^{\frac{1}{2}})}^{2} = {(- 4 x - 6)}^{2}$

Simplify the left side.

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

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

$2^{2} {({(6 - x)}^{\frac{1}{2}})}^{2} = {(- 4 x - 6)}^{2}$

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

$4 {({(6 - x)}^{\frac{1}{2}})}^{2} = {(- 4 x - 6)}^{2}$

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

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

$4 {(6 - x)}^{\frac{1}{2} \cdot 2} = {(- 4 x - 6)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$4 {(6 - x)}^{\frac{1}{2} \cdot 2} = {(- 4 x - 6)}^{2}$

Rewrite the expression.

$4 {(6 - x)}^{1} = {(- 4 x - 6)}^{2}$

Simplify.

$4 (6 - x) = {(- 4 x - 6)}^{2}$

Apply the distributive property.

$4 \cdot 6 + 4 (- x) = {(- 4 x - 6)}^{2}$

Multiply.

Multiply $4$ by $6$ .

$24 + 4 (- x) = {(- 4 x - 6)}^{2}$

Multiply $- 1$ by $4$ .

$24 - 4 x = {(- 4 x - 6)}^{2}$

Simplify the right side.

Simplify ${(- 4 x - 6)}^{2}$ .

Rewrite ${(- 4 x - 6)}^{2}$ as $(- 4 x - 6) (- 4 x - 6)$ .

$24 - 4 x = (- 4 x - 6) (- 4 x - 6)$

Expand $(- 4 x - 6) (- 4 x - 6)$ using the FOIL Method.

Apply the distributive property.

$24 - 4 x = - 4 x (- 4 x - 6) - 6 (- 4 x - 6)$

Apply the distributive property.

$24 - 4 x = - 4 x (- 4 x) - 4 x \cdot - 6 - 6 (- 4 x - 6)$

Apply the distributive property.

$24 - 4 x = - 4 x (- 4 x) - 4 x \cdot - 6 - 6 (- 4 x) - 6 \cdot - 6$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$24 - 4 x = - 4 \cdot - 4 x \cdot x - 4 x \cdot - 6 - 6 (- 4 x) - 6 \cdot - 6$

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

Move $x$ .

$24 - 4 x = - 4 \cdot - 4 (x \cdot x) - 4 x \cdot - 6 - 6 (- 4 x) - 6 \cdot - 6$

Multiply $x$ by $x$ .

$24 - 4 x = - 4 \cdot - 4 x^{2} - 4 x \cdot - 6 - 6 (- 4 x) - 6 \cdot - 6$

Multiply $- 4$ by $- 4$ .

$24 - 4 x = 16 x^{2} - 4 x \cdot - 6 - 6 (- 4 x) - 6 \cdot - 6$

Multiply $- 6$ by $- 4$ .

$24 - 4 x = 16 x^{2} + 24 x - 6 (- 4 x) - 6 \cdot - 6$

Multiply $- 4$ by $- 6$ .

$24 - 4 x = 16 x^{2} + 24 x + 24 x - 6 \cdot - 6$

Multiply $- 6$ by $- 6$ .

$24 - 4 x = 16 x^{2} + 24 x + 24 x + 36$

Add $24 x$ and $24 x$ .

$24 - 4 x = 16 x^{2} + 48 x + 36$

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.

$16 x^{2} + 48 x + 36 = 24 - 4 x$

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

Add $4 x$ to both sides of the equation.

$16 x^{2} + 48 x + 36 + 4 x = 24$

Add $48 x$ and $4 x$ .

$16 x^{2} + 52 x + 36 = 24$

Subtract $24$ from both sides of the equation.

$16 x^{2} + 52 x + 36 - 24 = 0$

Subtract $24$ from $36$ .

$16 x^{2} + 52 x + 12 = 0$

Factor the left side of the equation.

Factor $4$ out of $16 x^{2} + 52 x + 12$ .

Factor $4$ out of $16 x^{2}$ .

$4 (4 x^{2}) + 52 x + 12 = 0$

Factor $4$ out of $52 x$ .

$4 (4 x^{2}) + 4 (13 x) + 12 = 0$

Factor $4$ out of $12$ .

$4 (4 x^{2}) + 4 (13 x) + 4 (3) = 0$

Factor $4$ out of $4 (4 x^{2}) + 4 (13 x)$ .

$4 (4 x^{2} + 13 x) + 4 (3) = 0$

Factor $4$ out of $4 (4 x^{2} + 13 x) + 4 (3)$ .

$4 (4 x^{2} + 13 x + 3) = 0$

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 $a \cdot c = 4 \cdot 3 = 12$ and whose sum is $b = 13$ .

Factor $13$ out of $13 x$ .

$4 (4 x^{2} + 13 (x) + 3) = 0$

Rewrite $13$ as $1$ plus $12$

$4 (4 x^{2} + (1 + 12) x + 3) = 0$

Apply the distributive property.

$4 (4 x^{2} + 1 x + 12 x + 3) = 0$

Multiply $x$ by $1$ .

$4 (4 x^{2} + x + 12 x + 3) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$4 ((4 x^{2} + x) + 12 x + 3) = 0$

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

$4 (x (4 x + 1) + 3 (4 x + 1)) = 0$

Factor the polynomial by factoring out the greatest common factor, $4 x + 1$ .

$4 ((4 x + 1) (x + 3)) = 0$

Remove unnecessary parentheses.

$4 (4 x + 1) (x + 3) = 0$

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

$4 x + 1 = 0$

$x + 3 = 0$

Set $4 x + 1$ equal to $0$ and solve for $x$ .

Set $4 x + 1$ equal to $0$ .

$4 x + 1 = 0$

Solve $4 x + 1 = 0$ for $x$ .

Subtract $1$ from both sides of the equation.

$4 x = - 1$

Divide each term in $4 x = - 1$ by $4$ and simplify.

Divide each term in $4 x = - 1$ by $4$ .

$\frac{4 x}{4} = \frac{- 1}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 1}{4}$

Divide $x$ by $1$ .

$x = \frac{- 1}{4}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{1}{4}$

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

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

$x + 3 = 0$

Subtract $3$ from both sides of the equation.

$x = - 3$

The final solution is all the values that make $4 (4 x + 1) (x + 3) = 0$ true.

$x = - \frac{1}{4}, - 3$

Exclude the solutions that do not make $\sqrt{1 - 5 x} = 1 + \sqrt{6 - x}$ true.

$x = - 3$

Solve the Rational Equation for x square root of 1-5x=1+ square root of 6-x

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