Solve by Factoring fourth root of x^2+6x-2=0

$\sqrt[4]{x^{2} + 6 x} - 2 = 0$

Step 1

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

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

$\sqrt[4]{x \cdot x + 6 x} - 2 = 0$

Factor $x$ out of $6 x$ .

$\sqrt[4]{x \cdot x + x \cdot 6} - 2 = 0$

Factor $x$ out of $x \cdot x + x \cdot 6$ .

$\sqrt[4]{x (x + 6)} - 2 = 0$

Step 2

Add $2$ to both sides of the equation.

$\sqrt[4]{x (x + 6)} = 2$

Step 3

To remove the radical on the left side of the equation, raise both sides of the equation to the power of $4$ .

${\sqrt[4]{x (x + 6)}}^{4} = 2^{4}$

Step 4

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{x (x + 6)}$ as ${(x (x + 6))}^{\frac{1}{4}}$ .

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

Simplify the left side.

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

Multiply the exponents in ${({(x (x + 6))}^{\frac{1}{4}})}^{4}$ .

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

${(x (x + 6))}^{\frac{1}{4} \cdot 4} = 2^{4}$

Cancel the common factor of $4$ .

Cancel the common factor.

${(x (x + 6))}^{\frac{1}{4} \cdot 4} = 2^{4}$

Rewrite the expression.

${(x (x + 6))}^{1} = 2^{4}$

Apply the distributive property.

${(x \cdot x + x \cdot 6)}^{1} = 2^{4}$

Simplify the expression.

Multiply $x$ by $x$ .

${(x^{2} + x \cdot 6)}^{1} = 2^{4}$

Move $6$ to the left of $x$ .

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

Simplify the right side.

Raise $2$ to the power of $4$ .

$x^{2} + 6 x = 16$

Step 5

Solve for $x$ .

Subtract $16$ from both sides of the equation.

$x^{2} + 6 x - 16 = 0$

Factor $x^{2} + 6 x - 16$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 16$ and whose sum is $6$ .

$- 2, 8$

Write the factored form using these integers.

$(x - 2) (x + 8) = 0$

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

$x - 2 = 0$

$x + 8 = 0$

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 $x + 8$ equal to $0$ and solve for $x$ .

Set $x + 8$ equal to $0$ .

$x + 8 = 0$

Subtract $8$ from both sides of the equation.

$x = - 8$

The final solution is all the values that make $(x - 2) (x + 8) = 0$ true.

$x = 2, - 8$

Solve by Factoring fourth root of x^2+6x-2=0

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