Algebra Examples

$\sqrt{x} + 6 = x$

Step 1

Subtract $x$ from both sides of the equation.

$\sqrt{x} + 6 - x = 0$

Step 2

Factor the left side of the equation.

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Step 2.1

Factor $- 1$ out of $\sqrt{x} + 6 - x$ .

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Step 2.1.1

Reorder the expression.

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Step 2.1.1.1

Move $6$ .

$\sqrt{x} - x + 6 = 0$

Step 2.1.1.2

Reorder $\sqrt{x}$ and $- x$ .

$- x + \sqrt{x} + 6 = 0$

Step 2.1.2

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

$- (x) + \sqrt{x} + 6 = 0$

Step 2.1.3

Factor $- 1$ out of $\sqrt{x}$ .

$- (x) - 1 (- \sqrt{x}) + 6 = 0$

Step 2.1.4

Rewrite $6$ as $- 1 (- 6)$ .

$- (x) - 1 (- \sqrt{x}) - 1 \cdot - 6 = 0$

Step 2.1.5

Factor $- 1$ out of $- (x) - 1 (- \sqrt{x})$ .

$- (x - \sqrt{x}) - 1 \cdot - 6 = 0$

Step 2.1.6

Factor $- 1$ out of $- (x - \sqrt{x}) - 1 (- 6)$ .

$- (x - \sqrt{x} - 6) = 0$

Step 2.2

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

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

Step 2.3

Rewrite $x$ as ${(x^{\frac{1}{2}})}^{2}$ .

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

Step 2.4

Let $u = x^{\frac{1}{2}}$ . Substitute $u$ for all occurrences of $x^{\frac{1}{2}}$ .

$- (u^{2} - u - 6) = 0$

Step 2.5

Factor $u^{2} - u - 6$ using the AC method.

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Step 2.5.1

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 $- 6$ and whose sum is $- 1$ .

$- 3, 2$

Step 2.5.2

Write the factored form using these integers.

$- ((u - 3) (u + 2)) = 0$

Step 2.6

Factor.

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Step 2.6.1

Replace all occurrences of $u$ with $x^{\frac{1}{2}}$ .

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

Step 2.6.2

Remove unnecessary parentheses.

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

Step 3

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

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

$x^{\frac{1}{2}} + 2 = 0$

Step 4

Set $x^{\frac{1}{2}} - 3$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $x^{\frac{1}{2}} - 3$ equal to $0$ .

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

Step 4.2

Solve $x^{\frac{1}{2}} - 3 = 0$ for $x$ .

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Step 4.2.1

Add $3$ to both sides of the equation.

$x^{\frac{1}{2}} = 3$

Step 4.2.2

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

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

Step 4.2.3

Simplify the exponent.

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Step 4.2.3.1

Simplify the left side.

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Step 4.2.3.1.1

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

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Step 4.2.3.1.1.1

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

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Step 4.2.3.1.1.1.1

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

$x^{\frac{1}{2} \cdot 2} = 3^{2}$

Step 4.2.3.1.1.1.2

Cancel the common factor of $2$ .

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Step 4.2.3.1.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = 3^{2}$

Step 4.2.3.1.1.1.2.2

Rewrite the expression.

$x^{1} = 3^{2}$

Step 4.2.3.1.1.2

Simplify.

$x = 3^{2}$

Step 4.2.3.2

Simplify the right side.

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Step 4.2.3.2.1

Raise $3$ to the power of $2$ .

$x = 9$

Step 5

Set $x^{\frac{1}{2}} + 2$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $x^{\frac{1}{2}} + 2$ equal to $0$ .

$x^{\frac{1}{2}} + 2 = 0$

Step 5.2

Solve $x^{\frac{1}{2}} + 2 = 0$ for $x$ .

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Step 5.2.1

Subtract $2$ from both sides of the equation.

$x^{\frac{1}{2}} = - 2$

Step 5.2.2

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

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

Step 5.2.3

Simplify the exponent.

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Step 5.2.3.1

Simplify the left side.

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Step 5.2.3.1.1

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

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Step 5.2.3.1.1.1

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

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Step 5.2.3.1.1.1.1

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

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

Step 5.2.3.1.1.1.2

Cancel the common factor of $2$ .

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Step 5.2.3.1.1.1.2.1

Cancel the common factor.

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

Step 5.2.3.1.1.1.2.2

Rewrite the expression.

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

Step 5.2.3.1.1.2

Simplify.

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

Step 5.2.3.2

Simplify the right side.

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Step 5.2.3.2.1

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

$x = 4$

Step 6

The final solution is all the values that make $- (x^{\frac{1}{2}} - 3) (x^{\frac{1}{2}} + 2) = 0$ true.

$x = 9, 4$