Solve for x -x+6 = square root of x+4+3

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

Step 1

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

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

Step 2

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

Subtract $3$ from both sides of the equation.

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

Subtract $3$ from $6$ .

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

Step 3

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

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

Step 4

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

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

$x + 4 = (- x + 3) (- x + 3)$

Expand $(- x + 3) (- x + 3)$ using the FOIL Method.

Apply the distributive property.

$x + 4 = - x (- x + 3) + 3 (- x + 3)$

Apply the distributive property.

$x + 4 = - x (- x) - x \cdot 3 + 3 (- x + 3)$

Apply the distributive property.

$x + 4 = - x (- x) - x \cdot 3 + 3 (- x) + 3 \cdot 3$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$x + 4 = - 1 \cdot - 1 x \cdot x - x \cdot 3 + 3 (- x) + 3 \cdot 3$

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

Move $x$ .

$x + 4 = - 1 \cdot - 1 (x \cdot x) - x \cdot 3 + 3 (- x) + 3 \cdot 3$

Multiply $x$ by $x$ .

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

Multiply $- 1$ by $- 1$ .

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

Multiply $x^{2}$ by $1$ .

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

Multiply $3$ by $- 1$ .

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

Multiply $- 1$ by $3$ .

$x + 4 = x^{2} - 3 x - 3 x + 3 \cdot 3$

Multiply $3$ by $3$ .

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

Subtract $3 x$ from $- 3 x$ .

$x + 4 = x^{2} - 6 x + 9$

Step 5

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.

$x^{2} - 6 x + 9 = x + 4$

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

Subtract $x$ from both sides of the equation.

$x^{2} - 6 x + 9 - x = 4$

Subtract $x$ from $- 6 x$ .

$x^{2} - 7 x + 9 = 4$

Move all terms to the left side of the equation and simplify.

Subtract $4$ from both sides of the equation.

$x^{2} - 7 x + 9 - 4 = 0$

Subtract $4$ from $9$ .

$x^{2} - 7 x + 5 = 0$

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 1$ , $b = - 7$ , and $c = 5$ into the quadratic formula and solve for $x$ .

$\frac{7 \pm \sqrt{{(- 7)}^{2} - 4 \cdot (1 \cdot 5)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

Raise $- 7$ to the power of $2$ .

$x = \frac{7 \pm \sqrt{49 - 4 \cdot 1 \cdot 5}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 5$ .

Multiply $- 4$ by $1$ .

$x = \frac{7 \pm \sqrt{49 - 4 \cdot 5}}{2 \cdot 1}$

Multiply $- 4$ by $5$ .

$x = \frac{7 \pm \sqrt{49 - 20}}{2 \cdot 1}$

Subtract $20$ from $49$ .

$x = \frac{7 \pm \sqrt{29}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{7 \pm \sqrt{29}}{2}$

The final answer is the combination of both solutions.

$x = \frac{7 + \sqrt{29}}{2}, \frac{7 - \sqrt{29}}{2}$

Step 6

Exclude the solutions that do not make $- x + 6 = \sqrt{x + 4} + 3$ true.

$x = \frac{7 - \sqrt{29}}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \frac{7 - \sqrt{29}}{2}$

Decimal Form:

$x = 0.80741759 \dots$

Solve for x -x+6 = square root of x+4+3

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