Solve by Factoring square root of 3x-2+ square root of 11+x=-1

$\sqrt{3 x - 2} + \sqrt{11 + x} = - 1$

Step 1

Add $1$ to both sides of the equation.

$\sqrt{3 x - 2} + \sqrt{11 + x} + 1 = 0$

Step 2

Move all terms not containing $\sqrt{3 x - 2}$ to the right side of the equation.

Subtract $\sqrt{11 + x}$ from both sides of the equation.

$\sqrt{3 x - 2} + 1 = - \sqrt{11 + x}$

Subtract $1$ from both sides of the equation.

$\sqrt{3 x - 2} = - \sqrt{11 + x} - 1$

Step 3

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

${\sqrt{3 x - 2}}^{2} = {(- \sqrt{11 + x} - 1)}^{2}$

Step 4

Simplify each side of the equation.

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

${({(3 x - 2)}^{\frac{1}{2}})}^{2} = {(- \sqrt{11 + x} - 1)}^{2}$

Simplify the left side.

Simplify ${({(3 x - 2)}^{\frac{1}{2}})}^{2}$ .

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

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

${(3 x - 2)}^{\frac{1}{2} \cdot 2} = {(- \sqrt{11 + x} - 1)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(3 x - 2)}^{\frac{1}{2} \cdot 2} = {(- \sqrt{11 + x} - 1)}^{2}$

Rewrite the expression.

${(3 x - 2)}^{1} = {(- \sqrt{11 + x} - 1)}^{2}$

Simplify.

$3 x - 2 = {(- \sqrt{11 + x} - 1)}^{2}$

Simplify the right side.

Simplify ${(- \sqrt{11 + x} - 1)}^{2}$ .

Rewrite ${(- \sqrt{11 + x} - 1)}^{2}$ as $(- \sqrt{11 + x} - 1) (- \sqrt{11 + x} - 1)$ .

$3 x - 2 = (- \sqrt{11 + x} - 1) (- \sqrt{11 + x} - 1)$

Expand $(- \sqrt{11 + x} - 1) (- \sqrt{11 + x} - 1)$ using the FOIL Method.

Apply the distributive property.

$3 x - 2 = - \sqrt{11 + x} (- \sqrt{11 + x} - 1) - 1 (- \sqrt{11 + x} - 1)$

Apply the distributive property.

$3 x - 2 = - \sqrt{11 + x} (- \sqrt{11 + x}) - \sqrt{11 + x} \cdot - 1 - 1 (- \sqrt{11 + x} - 1)$

Apply the distributive property.

$3 x - 2 = - \sqrt{11 + x} (- \sqrt{11 + x}) - \sqrt{11 + x} \cdot - 1 - 1 (- \sqrt{11 + x}) - 1 \cdot - 1$

Simplify and combine like terms.

Simplify each term.

Multiply $- \sqrt{11 + x} (- \sqrt{11 + x})$ .

Multiply $- 1$ by $- 1$ .

$3 x - 2 = 1 \sqrt{11 + x} \sqrt{11 + x} - \sqrt{11 + x} \cdot - 1 - 1 (- \sqrt{11 + x}) - 1 \cdot - 1$

Multiply $\sqrt{11 + x}$ by $1$ .

$3 x - 2 = \sqrt{11 + x} \sqrt{11 + x} - \sqrt{11 + x} \cdot - 1 - 1 (- \sqrt{11 + x}) - 1 \cdot - 1$

Raise $\sqrt{11 + x}$ to the power of $1$ .

$3 x - 2 = {\sqrt{11 + x}}^{1} \sqrt{11 + x} - \sqrt{11 + x} \cdot - 1 - 1 (- \sqrt{11 + x}) - 1 \cdot - 1$

Raise $\sqrt{11 + x}$ to the power of $1$ .

$3 x - 2 = {\sqrt{11 + x}}^{1} {\sqrt{11 + x}}^{1} - \sqrt{11 + x} \cdot - 1 - 1 (- \sqrt{11 + x}) - 1 \cdot - 1$

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

$3 x - 2 = {\sqrt{11 + x}}^{1 + 1} - \sqrt{11 + x} \cdot - 1 - 1 (- \sqrt{11 + x}) - 1 \cdot - 1$

Add $1$ and $1$ .

$3 x - 2 = {\sqrt{11 + x}}^{2} - \sqrt{11 + x} \cdot - 1 - 1 (- \sqrt{11 + x}) - 1 \cdot - 1$

Rewrite ${\sqrt{11 + x}}^{2}$ as $11 + x$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11 + x}$ as ${(11 + x)}^{\frac{1}{2}}$ .

$3 x - 2 = {({(11 + x)}^{\frac{1}{2}})}^{2} - \sqrt{11 + x} \cdot - 1 - 1 (- \sqrt{11 + x}) - 1 \cdot - 1$

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

$3 x - 2 = {(11 + x)}^{\frac{1}{2} \cdot 2} - \sqrt{11 + x} \cdot - 1 - 1 (- \sqrt{11 + x}) - 1 \cdot - 1$

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

$3 x - 2 = {(11 + x)}^{\frac{2}{2}} - \sqrt{11 + x} \cdot - 1 - 1 (- \sqrt{11 + x}) - 1 \cdot - 1$

Cancel the common factor of $2$ .

Cancel the common factor.

$3 x - 2 = {(11 + x)}^{\frac{2}{2}} - \sqrt{11 + x} \cdot - 1 - 1 (- \sqrt{11 + x}) - 1 \cdot - 1$

Rewrite the expression.

$3 x - 2 = {(11 + x)}^{1} - \sqrt{11 + x} \cdot - 1 - 1 (- \sqrt{11 + x}) - 1 \cdot - 1$

Simplify.

$3 x - 2 = 11 + x - \sqrt{11 + x} \cdot - 1 - 1 (- \sqrt{11 + x}) - 1 \cdot - 1$

Multiply $- \sqrt{11 + x} \cdot - 1$ .

Multiply $- 1$ by $- 1$ .

$3 x - 2 = 11 + x + 1 \sqrt{11 + x} - 1 (- \sqrt{11 + x}) - 1 \cdot - 1$

Multiply $\sqrt{11 + x}$ by $1$ .

$3 x - 2 = 11 + x + \sqrt{11 + x} - 1 (- \sqrt{11 + x}) - 1 \cdot - 1$

Multiply $- 1 (- \sqrt{11 + x})$ .

Multiply $- 1$ by $- 1$ .

$3 x - 2 = 11 + x + \sqrt{11 + x} + 1 \sqrt{11 + x} - 1 \cdot - 1$

Multiply $\sqrt{11 + x}$ by $1$ .

$3 x - 2 = 11 + x + \sqrt{11 + x} + \sqrt{11 + x} - 1 \cdot - 1$

Multiply $- 1$ by $- 1$ .

$3 x - 2 = 11 + x + \sqrt{11 + x} + \sqrt{11 + x} + 1$

Add $11$ and $1$ .

$3 x - 2 = 12 + x + \sqrt{11 + x} + \sqrt{11 + x}$

Add $\sqrt{11 + x}$ and $\sqrt{11 + x}$ .

$3 x - 2 = 12 + x + 2 \sqrt{11 + x}$

Step 5

Solve for $2 \sqrt{11 + x}$ .

Rewrite the equation as $12 + x + 2 \sqrt{11 + x} = 3 x - 2$ .

$12 + x + 2 \sqrt{11 + x} = 3 x - 2$

Move all terms not containing $2 \sqrt{11 + x}$ to the right side of the equation.

Subtract $12$ from both sides of the equation.

$x + 2 \sqrt{11 + x} = 3 x - 2 - 12$

Subtract $x$ from both sides of the equation.

$2 \sqrt{11 + x} = 3 x - 2 - 12 - x$

Subtract $x$ from $3 x$ .

$2 \sqrt{11 + x} = 2 x - 2 - 12$

Subtract $12$ from $- 2$ .

$2 \sqrt{11 + x} = 2 x - 14$

Step 6

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

${(2 \sqrt{11 + x})}^{2} = {(2 x - 14)}^{2}$

Step 7

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11 + x}$ as ${(11 + x)}^{\frac{1}{2}}$ .

${(2 {(11 + x)}^{\frac{1}{2}})}^{2} = {(2 x - 14)}^{2}$

Simplify the left side.

Simplify ${(2 {(11 + x)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $2 {(11 + x)}^{\frac{1}{2}}$ .

$2^{2} {({(11 + x)}^{\frac{1}{2}})}^{2} = {(2 x - 14)}^{2}$

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

$4 {({(11 + x)}^{\frac{1}{2}})}^{2} = {(2 x - 14)}^{2}$

Multiply the exponents in ${({(11 + x)}^{\frac{1}{2}})}^{2}$ .

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

$4 {(11 + x)}^{\frac{1}{2} \cdot 2} = {(2 x - 14)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$4 {(11 + x)}^{\frac{1}{2} \cdot 2} = {(2 x - 14)}^{2}$

Rewrite the expression.

$4 {(11 + x)}^{1} = {(2 x - 14)}^{2}$

Simplify.

$4 (11 + x) = {(2 x - 14)}^{2}$

Apply the distributive property.

$4 \cdot 11 + 4 x = {(2 x - 14)}^{2}$

Multiply $4$ by $11$ .

$44 + 4 x = {(2 x - 14)}^{2}$

Simplify the right side.

Simplify ${(2 x - 14)}^{2}$ .

Rewrite ${(2 x - 14)}^{2}$ as $(2 x - 14) (2 x - 14)$ .

$44 + 4 x = (2 x - 14) (2 x - 14)$

Expand $(2 x - 14) (2 x - 14)$ using the FOIL Method.

Apply the distributive property.

$44 + 4 x = 2 x (2 x - 14) - 14 (2 x - 14)$

Apply the distributive property.

$44 + 4 x = 2 x (2 x) + 2 x \cdot - 14 - 14 (2 x - 14)$

Apply the distributive property.

$44 + 4 x = 2 x (2 x) + 2 x \cdot - 14 - 14 (2 x) - 14 \cdot - 14$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$44 + 4 x = 2 \cdot 2 x \cdot x + 2 x \cdot - 14 - 14 (2 x) - 14 \cdot - 14$

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

Move $x$ .

$44 + 4 x = 2 \cdot 2 (x \cdot x) + 2 x \cdot - 14 - 14 (2 x) - 14 \cdot - 14$

Multiply $x$ by $x$ .

$44 + 4 x = 2 \cdot 2 x^{2} + 2 x \cdot - 14 - 14 (2 x) - 14 \cdot - 14$

Multiply $2$ by $2$ .

$44 + 4 x = 4 x^{2} + 2 x \cdot - 14 - 14 (2 x) - 14 \cdot - 14$

Multiply $- 14$ by $2$ .

$44 + 4 x = 4 x^{2} - 28 x - 14 (2 x) - 14 \cdot - 14$

Multiply $2$ by $- 14$ .

$44 + 4 x = 4 x^{2} - 28 x - 28 x - 14 \cdot - 14$

Multiply $- 14$ by $- 14$ .

$44 + 4 x = 4 x^{2} - 28 x - 28 x + 196$

Subtract $28 x$ from $- 28 x$ .

$44 + 4 x = 4 x^{2} - 56 x + 196$

Step 8

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.

$4 x^{2} - 56 x + 196 = 44 + 4 x$

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

Subtract $4 x$ from both sides of the equation.

$4 x^{2} - 56 x + 196 - 4 x = 44$

Subtract $4 x$ from $- 56 x$ .

$4 x^{2} - 60 x + 196 = 44$

Subtract $44$ from both sides of the equation.

$4 x^{2} - 60 x + 196 - 44 = 0$

Subtract $44$ from $196$ .

$4 x^{2} - 60 x + 152 = 0$

Factor $4$ out of $4 x^{2} - 60 x + 152$ .

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

$4 (x^{2}) - 60 x + 152 = 0$

Factor $4$ out of $- 60 x$ .

$4 (x^{2}) + 4 (- 15 x) + 152 = 0$

Factor $4$ out of $152$ .

$4 x^{2} + 4 (- 15 x) + 4 \cdot 38 = 0$

Factor $4$ out of $4 x^{2} + 4 (- 15 x)$ .

$4 (x^{2} - 15 x) + 4 \cdot 38 = 0$

Factor $4$ out of $4 (x^{2} - 15 x) + 4 \cdot 38$ .

$4 (x^{2} - 15 x + 38) = 0$

Divide each term in $4 (x^{2} - 15 x + 38) = 0$ by $4$ and simplify.

Divide each term in $4 (x^{2} - 15 x + 38) = 0$ by $4$ .

$\frac{4 (x^{2} - 15 x + 38)}{4} = \frac{0}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 (x^{2} - 15 x + 38)}{4} = \frac{0}{4}$

Divide $x^{2} - 15 x + 38$ by $1$ .

$x^{2} - 15 x + 38 = \frac{0}{4}$

Simplify the right side.

Divide $0$ by $4$ .

$x^{2} - 15 x + 38 = 0$

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = 1$ , $b = - 15$ , and $c = 38$ into the quadratic formula and solve for $x$ .

$\frac{15 \pm \sqrt{{(- 15)}^{2} - 4 \cdot (1 \cdot 38)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

Raise $- 15$ to the power of $2$ .

$x = \frac{15 \pm \sqrt{225 - 4 \cdot 1 \cdot 38}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 38$ .

Multiply $- 4$ by $1$ .

$x = \frac{15 \pm \sqrt{225 - 4 \cdot 38}}{2 \cdot 1}$

Multiply $- 4$ by $38$ .

$x = \frac{15 \pm \sqrt{225 - 152}}{2 \cdot 1}$

Subtract $152$ from $225$ .

$x = \frac{15 \pm \sqrt{73}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{15 \pm \sqrt{73}}{2}$

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Simplify the numerator.

Raise $- 15$ to the power of $2$ .

$x = \frac{15 \pm \sqrt{225 - 4 \cdot 1 \cdot 38}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 38$ .

Multiply $- 4$ by $1$ .

$x = \frac{15 \pm \sqrt{225 - 4 \cdot 38}}{2 \cdot 1}$

Multiply $- 4$ by $38$ .

$x = \frac{15 \pm \sqrt{225 - 152}}{2 \cdot 1}$

Subtract $152$ from $225$ .

$x = \frac{15 \pm \sqrt{73}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{15 \pm \sqrt{73}}{2}$

Change the $\pm$ to $+$ .

$x = \frac{15 + \sqrt{73}}{2}$

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Simplify the numerator.

Raise $- 15$ to the power of $2$ .

$x = \frac{15 \pm \sqrt{225 - 4 \cdot 1 \cdot 38}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 38$ .

Multiply $- 4$ by $1$ .

$x = \frac{15 \pm \sqrt{225 - 4 \cdot 38}}{2 \cdot 1}$

Multiply $- 4$ by $38$ .

$x = \frac{15 \pm \sqrt{225 - 152}}{2 \cdot 1}$

Subtract $152$ from $225$ .

$x = \frac{15 \pm \sqrt{73}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{15 \pm \sqrt{73}}{2}$

Change the $\pm$ to $-$ .

$x = \frac{15 - \sqrt{73}}{2}$

The final answer is the combination of both solutions.

$x = \frac{15 + \sqrt{73}}{2}, \frac{15 - \sqrt{73}}{2}$

Step 9

Exclude the solutions that do not make $\sqrt{3 x - 2} + \sqrt{11 + x} + 1 = 0$ true.

$No solution$

Solve by Factoring square root of 3x-2+ square root of 11+x=-1

Related Questions

Support LCMGCF.com by disabling your adblocker.