Algebra Examples

$\sqrt{64 x} = x + 12$

Step 1

Move all the expressions to the left side of the equation.

Tap for more steps...

Step 1.1

Subtract $x$ from both sides of the equation.

$\sqrt{64 x} - x = 12$

Step 1.2

Subtract $12$ from both sides of the equation.

$\sqrt{64 x} - x - 12 = 0$

Step 2

Simplify each term.

Tap for more steps...

Step 2.1

Rewrite $64$ as $8^{2}$ .

$\sqrt{8^{2} x} - x - 12 = 0$

Step 2.2

Pull terms out from under the radical.

$8 \sqrt{x} - x - 12 = 0$

Step 3

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

$8 x^{\frac{1}{2}} - x - 12 = 0$

Step 4

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

$8 x^{\frac{1}{2}} - {(x^{\frac{1}{2}})}^{2} - 12 = 0$

Step 5

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

$8 u - u^{2} - 12 = 0$

Step 6

Factor by grouping.

Tap for more steps...

Step 6.1

Reorder terms.

$- u^{2} + 8 u - 12 = 0$

Step 6.2

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 $a \cdot c = - 1 \cdot - 12 = 12$ and whose sum is $b = 8$ .

Tap for more steps...

Step 6.2.1

Factor $8$ out of $8 u$ .

$- u^{2} + 8 (u) - 12 = 0$

Step 6.2.2

Rewrite $8$ as $2$ plus $6$

$- u^{2} + (2 + 6) u - 12 = 0$

Step 6.2.3

Apply the distributive property.

$- u^{2} + 2 u + 6 u - 12 = 0$

Step 6.3

Factor out the greatest common factor from each group.

Tap for more steps...

Step 6.3.1

Group the first two terms and the last two terms.

$(- u^{2} + 2 u) + 6 u - 12 = 0$

Step 6.3.2

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

$u (- u + 2) - 6 (- u + 2) = 0$

Step 6.4

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

$(- u + 2) (u - 6) = 0$

Step 7

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

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

Step 8

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

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

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

Step 9

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

Tap for more steps...

Step 9.1

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

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

Step 9.2

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

Tap for more steps...

Step 9.2.1

Subtract $2$ from both sides of the equation.

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

Step 9.2.2

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 9.2.3

Simplify the exponent.

Tap for more steps...

Step 9.2.3.1

Simplify the left side.

Tap for more steps...

Step 9.2.3.1.1

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

Tap for more steps...

Step 9.2.3.1.1.1

Apply the product rule to $- x^{\frac{1}{2}}$ .

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

Step 9.2.3.1.1.2

Raise $- 1$ to the power of $2$ .

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

Step 9.2.3.1.1.3

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

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

Step 9.2.3.1.1.4

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

Tap for more steps...

Step 9.2.3.1.1.4.1

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

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

Step 9.2.3.1.1.4.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 9.2.3.1.1.4.2.1

Cancel the common factor.

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

Step 9.2.3.1.1.4.2.2

Rewrite the expression.

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

Step 9.2.3.1.1.5

Simplify.

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

Step 9.2.3.2

Simplify the right side.

Tap for more steps...

Step 9.2.3.2.1

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

$x = 4$

Step 10

Set $x^{\frac{1}{2}} - 6$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 10.1

Set $x^{\frac{1}{2}} - 6$ equal to $0$ .

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

Step 10.2

Solve $x^{\frac{1}{2}} - 6 = 0$ for $x$ .

Tap for more steps...

Step 10.2.1

Add $6$ to both sides of the equation.

$x^{\frac{1}{2}} = 6$

Step 10.2.2

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

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

Step 10.2.3

Simplify the exponent.

Tap for more steps...

Step 10.2.3.1

Simplify the left side.

Tap for more steps...

Step 10.2.3.1.1

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

Tap for more steps...

Step 10.2.3.1.1.1

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

Tap for more steps...

Step 10.2.3.1.1.1.1

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

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

Step 10.2.3.1.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 10.2.3.1.1.1.2.1

Cancel the common factor.

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

Step 10.2.3.1.1.1.2.2

Rewrite the expression.

$x^{1} = 6^{2}$

Step 10.2.3.1.1.2

Simplify.

$x = 6^{2}$

Step 10.2.3.2

Simplify the right side.

Tap for more steps...

Step 10.2.3.2.1

Raise $6$ to the power of $2$ .

$x = 36$

Step 11

The final solution is all the values that make $(- x^{\frac{1}{2}} + 2) (x^{\frac{1}{2}} - 6) = 0$ true.

$x = 4, 36$