Solve Using the Square Root Property 3(x-2)^2-6=34

$3 {(x - 2)}^{2} - 6 = 34$

Step 1

Add $6$ to both sides of the equation.

$3 {(x - 2)}^{2} = 34 + 6$

Step 2

Add $34$ and $6$ .

$3 {(x - 2)}^{2} = 40$

Step 3

Divide each term in $3 {(x - 2)}^{2} = 40$ by $3$ and simplify.

Divide each term in $3 {(x - 2)}^{2} = 40$ by $3$ .

$\frac{3 {(x - 2)}^{2}}{3} = \frac{40}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 {(x - 2)}^{2}}{3} = \frac{40}{3}$

Divide ${(x - 2)}^{2}$ by $1$ .

${(x - 2)}^{2} = \frac{40}{3}$

Step 4

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x - 2 = \pm \sqrt{\frac{40}{3}}$

Step 5

Simplify $\pm \sqrt{\frac{40}{3}}$ .

Rewrite $\sqrt{\frac{40}{3}}$ as $\frac{\sqrt{40}}{\sqrt{3}}$ .

$x - 2 = \pm \frac{\sqrt{40}}{\sqrt{3}}$

Simplify the numerator.

Rewrite $40$ as $2^{2} \cdot 10$ .

Factor $4$ out of $40$ .

$x - 2 = \pm \frac{\sqrt{4 (10)}}{\sqrt{3}}$

Rewrite $4$ as $2^{2}$ .

$x - 2 = \pm \frac{\sqrt{2^{2} \cdot 10}}{\sqrt{3}}$

Pull terms out from under the radical.

$x - 2 = \pm \frac{2 \sqrt{10}}{\sqrt{3}}$

Multiply $\frac{2 \sqrt{10}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x - 2 = \pm \frac{2 \sqrt{10}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Combine and simplify the denominator.

Multiply $\frac{2 \sqrt{10}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x - 2 = \pm \frac{2 \sqrt{10} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

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

$x - 2 = \pm \frac{2 \sqrt{10} \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

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

$x - 2 = \pm \frac{2 \sqrt{10} \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

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

$x - 2 = \pm \frac{2 \sqrt{10} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Add $1$ and $1$ .

$x - 2 = \pm \frac{2 \sqrt{10} \sqrt{3}}{{\sqrt{3}}^{2}}$

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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

$x - 2 = \pm \frac{2 \sqrt{10} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

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

$x - 2 = \pm \frac{2 \sqrt{10} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

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

$x - 2 = \pm \frac{2 \sqrt{10} \sqrt{3}}{3^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x - 2 = \pm \frac{2 \sqrt{10} \sqrt{3}}{3^{\frac{2}{2}}}$

Rewrite the expression.

$x - 2 = \pm \frac{2 \sqrt{10} \sqrt{3}}{3^{1}}$

Evaluate the exponent.

$x - 2 = \pm \frac{2 \sqrt{10} \sqrt{3}}{3}$

Simplify the numerator.

Combine using the product rule for radicals.

$x - 2 = \pm \frac{2 \sqrt{3 \cdot 10}}{3}$

Multiply $3$ by $10$ .

$x - 2 = \pm \frac{2 \sqrt{30}}{3}$

Step 6

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x - 2 = \frac{2 \sqrt{30}}{3}$

Add $2$ to both sides of the equation.

$x = \frac{2 \sqrt{30}}{3} + 2$

Next, use the negative value of the $\pm$ to find the second solution.

$x - 2 = - \frac{2 \sqrt{30}}{3}$

Add $2$ to both sides of the equation.

$x = - \frac{2 \sqrt{30}}{3} + 2$

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{2 \sqrt{30}}{3} + 2, - \frac{2 \sqrt{30}}{3} + 2$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \frac{2 \sqrt{30}}{3} + 2, - \frac{2 \sqrt{30}}{3} + 2$

Decimal Form:

$x = 5.65148371 \dots, - 1.65148371 \dots$

Solve Using the Square Root Property 3(x-2)^2-6=34

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