Algebra Examples

$\sqrt{x^{2} + 6 x - 9} = \sqrt{2 x^{2}}$

Step 1

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

${\sqrt{x^{2} + 6 x - 9}}^{2} = {\sqrt{2 x^{2}}}^{2}$

Step 2

Simplify each side of the equation.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} + 6 x - 9}$ as ${(x^{2} + 6 x - 9)}^{\frac{1}{2}}$ .

${({(x^{2} + 6 x - 9)}^{\frac{1}{2}})}^{2} = {\sqrt{2 x^{2}}}^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(x^{2} + 6 x - 9)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(x^{2} + 6 x - 9)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

${(x^{2} + 6 x - 9)}^{\frac{1}{2} \cdot 2} = {\sqrt{2 x^{2}}}^{2}$

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

${(x^{2} + 6 x - 9)}^{\frac{1}{2} \cdot 2} = {\sqrt{2 x^{2}}}^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(x^{2} + 6 x - 9)}^{1} = {\sqrt{2 x^{2}}}^{2}$

Step 2.2.1.2

Simplify.

$x^{2} + 6 x - 9 = {\sqrt{2 x^{2}}}^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify ${\sqrt{2 x^{2}}}^{2}$ .

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Step 2.3.1.1

Reorder $2$ and $x^{2}$ .

$x^{2} + 6 x - 9 = {\sqrt{x^{2} \cdot 2}}^{2}$

Step 2.3.1.2

Pull terms out from under the radical.

$x^{2} + 6 x - 9 = {(x \sqrt{2})}^{2}$

Step 2.3.1.3

Apply the product rule to $x \sqrt{2}$ .

$x^{2} + 6 x - 9 = x^{2} {\sqrt{2}}^{2}$

Step 2.3.1.4

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

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Step 2.3.1.4.1

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

$x^{2} + 6 x - 9 = x^{2} {(2^{\frac{1}{2}})}^{2}$

Step 2.3.1.4.2

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

$x^{2} + 6 x - 9 = x^{2} \cdot 2^{\frac{1}{2} \cdot 2}$

Step 2.3.1.4.3

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

$x^{2} + 6 x - 9 = x^{2} \cdot 2^{\frac{2}{2}}$

Step 2.3.1.4.4

Cancel the common factor of $2$ .

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Step 2.3.1.4.4.1

Cancel the common factor.

$x^{2} + 6 x - 9 = x^{2} \cdot 2^{\frac{2}{2}}$

Step 2.3.1.4.4.2

Rewrite the expression.

$x^{2} + 6 x - 9 = x^{2} \cdot 2^{1}$

Step 2.3.1.4.5

Evaluate the exponent.

$x^{2} + 6 x - 9 = x^{2} \cdot 2$

Step 2.3.1.5

Move $2$ to the left of $x^{2}$ .

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

Step 3

Solve for $x$ .

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Step 3.1

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

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Step 3.1.1

Subtract $2 x^{2}$ from both sides of the equation.

$x^{2} + 6 x - 9 - 2 x^{2} = 0$

Step 3.1.2

Subtract $2 x^{2}$ from $x^{2}$ .

$- x^{2} + 6 x - 9 = 0$

Step 3.2

Factor the left side of the equation.

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Step 3.2.1

Factor $- 1$ out of $- x^{2} + 6 x - 9$ .

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Step 3.2.1.1

Factor $- 1$ out of $- x^{2}$ .

$- (x^{2}) + 6 x - 9 = 0$

Step 3.2.1.2

Factor $- 1$ out of $6 x$ .

$- (x^{2}) - (- 6 x) - 9 = 0$

Step 3.2.1.3

Rewrite $- 9$ as $- 1 (9)$ .

$- (x^{2}) - (- 6 x) - 1 \cdot 9 = 0$

Step 3.2.1.4

Factor $- 1$ out of $- (x^{2}) - (- 6 x)$ .

$- (x^{2} - 6 x) - 1 \cdot 9 = 0$

Step 3.2.1.5

Factor $- 1$ out of $- (x^{2} - 6 x) - 1 (9)$ .

$- (x^{2} - 6 x + 9) = 0$

Step 3.2.2

Factor using the perfect square rule.

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Step 3.2.2.1

Rewrite $9$ as $3^{2}$ .

$- (x^{2} - 6 x + 3^{2}) = 0$

Step 3.2.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$6 x = 2 \cdot x \cdot 3$

Step 3.2.2.3

Rewrite the polynomial.

$- (x^{2} - 2 \cdot x \cdot 3 + 3^{2}) = 0$

Step 3.2.2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 3$ .

$- {(x - 3)}^{2} = 0$

Step 3.3

Divide each term in $- {(x - 3)}^{2} = 0$ by $- 1$ and simplify.

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Step 3.3.1

Divide each term in $- {(x - 3)}^{2} = 0$ by $- 1$ .

$\frac{- {(x - 3)}^{2}}{- 1} = \frac{0}{- 1}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Dividing two negative values results in a positive value.

$\frac{{(x - 3)}^{2}}{1} = \frac{0}{- 1}$

Step 3.3.2.2

Divide ${(x - 3)}^{2}$ by $1$ .

${(x - 3)}^{2} = \frac{0}{- 1}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Divide $0$ by $- 1$ .

${(x - 3)}^{2} = 0$

Step 3.4

Set the $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 3.5

Add $3$ to both sides of the equation.

$x = 3$