Solve for x square root of x+2=2x-6

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

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

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

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

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

$x + 2 = (2 x - 6) (2 x - 6)$

Expand $(2 x - 6) (2 x - 6)$ using the FOIL Method.

Apply the distributive property.

$x + 2 = 2 x (2 x - 6) - 6 (2 x - 6)$

Apply the distributive property.

$x + 2 = 2 x (2 x) + 2 x \cdot - 6 - 6 (2 x - 6)$

Apply the distributive property.

$x + 2 = 2 x (2 x) + 2 x \cdot - 6 - 6 (2 x) - 6 \cdot - 6$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$x + 2 = 2 \cdot 2 x \cdot x + 2 x \cdot - 6 - 6 (2 x) - 6 \cdot - 6$

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

Move $x$ .

$x + 2 = 2 \cdot 2 (x \cdot x) + 2 x \cdot - 6 - 6 (2 x) - 6 \cdot - 6$

Multiply $x$ by $x$ .

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

Multiply $2$ by $2$ .

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

Multiply $- 6$ by $2$ .

$x + 2 = 4 x^{2} - 12 x - 6 (2 x) - 6 \cdot - 6$

Multiply $2$ by $- 6$ .

$x + 2 = 4 x^{2} - 12 x - 12 x - 6 \cdot - 6$

Multiply $- 6$ by $- 6$ .

$x + 2 = 4 x^{2} - 12 x - 12 x + 36$

Subtract $12 x$ from $- 12 x$ .

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

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.

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

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

Subtract $x$ from both sides of the equation.

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

Subtract $x$ from $- 24 x$ .

$4 x^{2} - 25 x + 36 = 2$

Subtract $2$ from both sides of the equation.

$4 x^{2} - 25 x + 36 - 2 = 0$

Subtract $2$ from $36$ .

$4 x^{2} - 25 x + 34 = 0$

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 $a \cdot c = 4 \cdot 34 = 136$ and whose sum is $b = - 25$ .

Factor $- 25$ out of $- 25 x$ .

$4 x^{2} - 25 x + 34 = 0$

Rewrite $- 25$ as $- 8$ plus $- 17$

$4 x^{2} + (- 8 - 17) x + 34 = 0$

Apply the distributive property.

$4 x^{2} - 8 x - 17 x + 34 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(4 x^{2} - 8 x) - 17 x + 34 = 0$

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

$4 x (x - 2) - 17 (x - 2) = 0$

Factor the polynomial by factoring out the greatest common factor, $x - 2$ .

$(x - 2) (4 x - 17) = 0$

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

$x - 2 = 0$

$4 x - 17 = 0$

Set $x - 2$ equal to $0$ and solve for $x$ .

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

Set $4 x - 17$ equal to $0$ and solve for $x$ .

Set $4 x - 17$ equal to $0$ .

$4 x - 17 = 0$

Solve $4 x - 17 = 0$ for $x$ .

Add $17$ to both sides of the equation.

$4 x = 17$

Divide each term in $4 x = 17$ by $4$ and simplify.

Divide each term in $4 x = 17$ by $4$ .

$\frac{4 x}{4} = \frac{17}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{17}{4}$

Divide $x$ by $1$ .

$x = \frac{17}{4}$

The final solution is all the values that make $(x - 2) (4 x - 17) = 0$ true.

$x = 2, \frac{17}{4}$

Exclude the solutions that do not make $\sqrt{x + 2} = 2 x - 6$ true.

$x = \frac{17}{4}$

The result can be shown in multiple forms.

Exact Form:

$x = \frac{17}{4}$

Decimal Form:

$x = 4.25$

Mixed Number Form:

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

Solve for x square root of x+2=2x-6

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