Solve for x square root of -x+5+ square root of 3x+4=3

$\sqrt{- x + 5} + \sqrt{3 x + 4} = 3$

Subtract $\sqrt{3 x + 4}$ from both sides of the equation.

$\sqrt{- x + 5} = 3 - \sqrt{3 x + 4}$

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

${\sqrt{- x + 5}}^{2} = {(3 - \sqrt{3 x + 4})}^{2}$

Simplify each side of the equation.

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

${({(- x + 5)}^{\frac{1}{2}})}^{2} = {(3 - \sqrt{3 x + 4})}^{2}$

Simplify the left side.

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

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

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

${(- x + 5)}^{\frac{1}{2} \cdot 2} = {(3 - \sqrt{3 x + 4})}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(- x + 5)}^{\frac{1}{2} \cdot 2} = {(3 - \sqrt{3 x + 4})}^{2}$

Rewrite the expression.

${(- x + 5)}^{1} = {(3 - \sqrt{3 x + 4})}^{2}$

Simplify.

$- x + 5 = {(3 - \sqrt{3 x + 4})}^{2}$

Simplify the right side.

Simplify ${(3 - \sqrt{3 x + 4})}^{2}$ .

Rewrite ${(3 - \sqrt{3 x + 4})}^{2}$ as $(3 - \sqrt{3 x + 4}) (3 - \sqrt{3 x + 4})$ .

$- x + 5 = (3 - \sqrt{3 x + 4}) (3 - \sqrt{3 x + 4})$

Expand $(3 - \sqrt{3 x + 4}) (3 - \sqrt{3 x + 4})$ using the FOIL Method.

Apply the distributive property.

$- x + 5 = 3 (3 - \sqrt{3 x + 4}) - \sqrt{3 x + 4} (3 - \sqrt{3 x + 4})$

Apply the distributive property.

$- x + 5 = 3 \cdot 3 + 3 (- \sqrt{3 x + 4}) - \sqrt{3 x + 4} (3 - \sqrt{3 x + 4})$

Apply the distributive property.

$- x + 5 = 3 \cdot 3 + 3 (- \sqrt{3 x + 4}) - \sqrt{3 x + 4} \cdot 3 - \sqrt{3 x + 4} (- \sqrt{3 x + 4})$

Simplify and combine like terms.

Simplify each term.

Multiply $3$ by $3$ .

$- x + 5 = 9 + 3 (- \sqrt{3 x + 4}) - \sqrt{3 x + 4} \cdot 3 - \sqrt{3 x + 4} (- \sqrt{3 x + 4})$

Multiply $- 1$ by $3$ .

$- x + 5 = 9 - 3 \sqrt{3 x + 4} - \sqrt{3 x + 4} \cdot 3 - \sqrt{3 x + 4} (- \sqrt{3 x + 4})$

Multiply $3$ by $- 1$ .

$- x + 5 = 9 - 3 \sqrt{3 x + 4} - 3 \sqrt{3 x + 4} - \sqrt{3 x + 4} (- \sqrt{3 x + 4})$

Multiply $- \sqrt{3 x + 4} (- \sqrt{3 x + 4})$ .

Multiply $- 1$ by $- 1$ .

$- x + 5 = 9 - 3 \sqrt{3 x + 4} - 3 \sqrt{3 x + 4} + 1 \sqrt{3 x + 4} \sqrt{3 x + 4}$

Multiply $\sqrt{3 x + 4}$ by $1$ .

$- x + 5 = 9 - 3 \sqrt{3 x + 4} - 3 \sqrt{3 x + 4} + \sqrt{3 x + 4} \sqrt{3 x + 4}$

Raise $\sqrt{3 x + 4}$ to the power of $1$ .

$- x + 5 = 9 - 3 \sqrt{3 x + 4} - 3 \sqrt{3 x + 4} + {\sqrt{3 x + 4}}^{1} \sqrt{3 x + 4}$

Raise $\sqrt{3 x + 4}$ to the power of $1$ .

$- x + 5 = 9 - 3 \sqrt{3 x + 4} - 3 \sqrt{3 x + 4} + {\sqrt{3 x + 4}}^{1} {\sqrt{3 x + 4}}^{1}$

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

$- x + 5 = 9 - 3 \sqrt{3 x + 4} - 3 \sqrt{3 x + 4} + {\sqrt{3 x + 4}}^{1 + 1}$

Add $1$ and $1$ .

$- x + 5 = 9 - 3 \sqrt{3 x + 4} - 3 \sqrt{3 x + 4} + {\sqrt{3 x + 4}}^{2}$

Rewrite ${\sqrt{3 x + 4}}^{2}$ as $3 x + 4$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 x + 4}$ as ${(3 x + 4)}^{\frac{1}{2}}$ .

$- x + 5 = 9 - 3 \sqrt{3 x + 4} - 3 \sqrt{3 x + 4} + {({(3 x + 4)}^{\frac{1}{2}})}^{2}$

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

$- x + 5 = 9 - 3 \sqrt{3 x + 4} - 3 \sqrt{3 x + 4} + {(3 x + 4)}^{\frac{1}{2} \cdot 2}$

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

$- x + 5 = 9 - 3 \sqrt{3 x + 4} - 3 \sqrt{3 x + 4} + {(3 x + 4)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$- x + 5 = 9 - 3 \sqrt{3 x + 4} - 3 \sqrt{3 x + 4} + {(3 x + 4)}^{\frac{2}{2}}$

Rewrite the expression.

$- x + 5 = 9 - 3 \sqrt{3 x + 4} - 3 \sqrt{3 x + 4} + {(3 x + 4)}^{1}$

Simplify.

$- x + 5 = 9 - 3 \sqrt{3 x + 4} - 3 \sqrt{3 x + 4} + 3 x + 4$

Add $9$ and $4$ .

$- x + 5 = 13 - 3 \sqrt{3 x + 4} - 3 \sqrt{3 x + 4} + 3 x$

Subtract $3 \sqrt{3 x + 4}$ from $- 3 \sqrt{3 x + 4}$ .

$- x + 5 = 13 - 6 \sqrt{3 x + 4} + 3 x$

Solve for $- 6 \sqrt{3 x + 4}$ .

Rewrite the equation as $13 - 6 \sqrt{3 x + 4} + 3 x = - x + 5$ .

$13 - 6 \sqrt{3 x + 4} + 3 x = - x + 5$

Move all terms not containing $- 6 \sqrt{3 x + 4}$ to the right side of the equation.

Subtract $13$ from both sides of the equation.

$- 6 \sqrt{3 x + 4} + 3 x = - x + 5 - 13$

Subtract $3 x$ from both sides of the equation.

$- 6 \sqrt{3 x + 4} = - x + 5 - 13 - 3 x$

Subtract $3 x$ from $- x$ .

$- 6 \sqrt{3 x + 4} = - 4 x + 5 - 13$

Subtract $13$ from $5$ .

$- 6 \sqrt{3 x + 4} = - 4 x - 8$

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

${(- 6 \sqrt{3 x + 4})}^{2} = {(- 4 x - 8)}^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 x + 4}$ as ${(3 x + 4)}^{\frac{1}{2}}$ .

${(- 6 {(3 x + 4)}^{\frac{1}{2}})}^{2} = {(- 4 x - 8)}^{2}$

Simplify the left side.

Simplify ${(- 6 {(3 x + 4)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $- 6 {(3 x + 4)}^{\frac{1}{2}}$ .

${(- 6)}^{2} {({(3 x + 4)}^{\frac{1}{2}})}^{2} = {(- 4 x - 8)}^{2}$

Raise $- 6$ to the power of $2$ .

$36 {({(3 x + 4)}^{\frac{1}{2}})}^{2} = {(- 4 x - 8)}^{2}$

Multiply the exponents in ${({(3 x + 4)}^{\frac{1}{2}})}^{2}$ .

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

$36 {(3 x + 4)}^{\frac{1}{2} \cdot 2} = {(- 4 x - 8)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$36 {(3 x + 4)}^{\frac{1}{2} \cdot 2} = {(- 4 x - 8)}^{2}$

Rewrite the expression.

$36 {(3 x + 4)}^{1} = {(- 4 x - 8)}^{2}$

Simplify.

$36 (3 x + 4) = {(- 4 x - 8)}^{2}$

Apply the distributive property.

$36 (3 x) + 36 \cdot 4 = {(- 4 x - 8)}^{2}$

Multiply.

Multiply $3$ by $36$ .

$108 x + 36 \cdot 4 = {(- 4 x - 8)}^{2}$

Multiply $36$ by $4$ .

$108 x + 144 = {(- 4 x - 8)}^{2}$

Simplify the right side.

Simplify ${(- 4 x - 8)}^{2}$ .

Rewrite ${(- 4 x - 8)}^{2}$ as $(- 4 x - 8) (- 4 x - 8)$ .

$108 x + 144 = (- 4 x - 8) (- 4 x - 8)$

Expand $(- 4 x - 8) (- 4 x - 8)$ using the FOIL Method.

Apply the distributive property.

$108 x + 144 = - 4 x (- 4 x - 8) - 8 (- 4 x - 8)$

Apply the distributive property.

$108 x + 144 = - 4 x (- 4 x) - 4 x \cdot - 8 - 8 (- 4 x - 8)$

Apply the distributive property.

$108 x + 144 = - 4 x (- 4 x) - 4 x \cdot - 8 - 8 (- 4 x) - 8 \cdot - 8$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$108 x + 144 = - 4 \cdot - 4 x \cdot x - 4 x \cdot - 8 - 8 (- 4 x) - 8 \cdot - 8$

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

Move $x$ .

$108 x + 144 = - 4 \cdot - 4 (x \cdot x) - 4 x \cdot - 8 - 8 (- 4 x) - 8 \cdot - 8$

Multiply $x$ by $x$ .

$108 x + 144 = - 4 \cdot - 4 x^{2} - 4 x \cdot - 8 - 8 (- 4 x) - 8 \cdot - 8$

Multiply $- 4$ by $- 4$ .

$108 x + 144 = 16 x^{2} - 4 x \cdot - 8 - 8 (- 4 x) - 8 \cdot - 8$

Multiply $- 8$ by $- 4$ .

$108 x + 144 = 16 x^{2} + 32 x - 8 (- 4 x) - 8 \cdot - 8$

Multiply $- 4$ by $- 8$ .

$108 x + 144 = 16 x^{2} + 32 x + 32 x - 8 \cdot - 8$

Multiply $- 8$ by $- 8$ .

$108 x + 144 = 16 x^{2} + 32 x + 32 x + 64$

Add $32 x$ and $32 x$ .

$108 x + 144 = 16 x^{2} + 64 x + 64$

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.

$16 x^{2} + 64 x + 64 = 108 x + 144$

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

Subtract $108 x$ from both sides of the equation.

$16 x^{2} + 64 x + 64 - 108 x = 144$

Subtract $108 x$ from $64 x$ .

$16 x^{2} - 44 x + 64 = 144$

Subtract $144$ from both sides of the equation.

$16 x^{2} - 44 x + 64 - 144 = 0$

Subtract $144$ from $64$ .

$16 x^{2} - 44 x - 80 = 0$

Factor the left side of the equation.

Factor $4$ out of $16 x^{2} - 44 x - 80$ .

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

$4 (4 x^{2}) - 44 x - 80 = 0$

Factor $4$ out of $- 44 x$ .

$4 (4 x^{2}) + 4 (- 11 x) - 80 = 0$

Factor $4$ out of $- 80$ .

$4 (4 x^{2}) + 4 (- 11 x) + 4 (- 20) = 0$

Factor $4$ out of $4 (4 x^{2}) + 4 (- 11 x)$ .

$4 (4 x^{2} - 11 x) + 4 (- 20) = 0$

Factor $4$ out of $4 (4 x^{2} - 11 x) + 4 (- 20)$ .

$4 (4 x^{2} - 11 x - 20) = 0$

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 $a \cdot c = 4 \cdot - 20 = - 80$ and whose sum is $b = - 11$ .

Factor $- 11$ out of $- 11 x$ .

$4 (4 x^{2} - 11 x - 20) = 0$

Rewrite $- 11$ as $5$ plus $- 16$

$4 (4 x^{2} + (5 - 16) x - 20) = 0$

Apply the distributive property.

$4 (4 x^{2} + 5 x - 16 x - 20) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$4 ((4 x^{2} + 5 x) - 16 x - 20) = 0$

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

$4 (x (4 x + 5) - 4 (4 x + 5)) = 0$

Factor the polynomial by factoring out the greatest common factor, $4 x + 5$ .

$4 ((4 x + 5) (x - 4)) = 0$

Remove unnecessary parentheses.

$4 (4 x + 5) (x - 4) = 0$

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

$4 x + 5 = 0$

$x - 4 = 0$

Set $4 x + 5$ equal to $0$ and solve for $x$ .

Set $4 x + 5$ equal to $0$ .

$4 x + 5 = 0$

Solve $4 x + 5 = 0$ for $x$ .

Subtract $5$ from both sides of the equation.

$4 x = - 5$

Divide each term in $4 x = - 5$ by $4$ and simplify.

Divide each term in $4 x = - 5$ by $4$ .

$\frac{4 x}{4} = \frac{- 5}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 5}{4}$

Divide $x$ by $1$ .

$x = \frac{- 5}{4}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{5}{4}$

Set $x - 4$ equal to $0$ and solve for $x$ .

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Add $4$ to both sides of the equation.

$x = 4$

The final solution is all the values that make $4 (4 x + 5) (x - 4) = 0$ true.

$x = - \frac{5}{4}, 4$

Exclude the solutions that do not make $\sqrt{- x + 5} + \sqrt{3 x + 4} = 3$ true.

$x = - \frac{5}{4}$

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{5}{4}$

Decimal Form:

$x = - 1.25$

Mixed Number Form:

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

Solve for x square root of -x+5+ square root of 3x+4=3

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