Algebra Examples

$x - 5 \sqrt{x} + 6 = 0$

Step 1

Move all terms not containing $- 5 \sqrt{x}$ to the right side of the equation.

Tap for more steps...

Step 1.1

Subtract $x$ from both sides of the equation.

$- 5 \sqrt{x} + 6 = - x$

Step 1.2

Subtract $6$ from both sides of the equation.

$- 5 \sqrt{x} = - x - 6$

Step 2

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

${(- 5 \sqrt{x})}^{2} = {(- x - 6)}^{2}$

Step 3

Simplify each side of the equation.

Tap for more steps...

Step 3.1

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

${(- 5 x^{\frac{1}{2}})}^{2} = {(- x - 6)}^{2}$

Step 3.2

Simplify the left side.

Tap for more steps...

Step 3.2.1

Simplify ${(- 5 x^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 3.2.1.1

Apply the product rule to $- 5 x^{\frac{1}{2}}$ .

${(- 5)}^{2} {(x^{\frac{1}{2}})}^{2} = {(- x - 6)}^{2}$

Step 3.2.1.2

Raise $- 5$ to the power of $2$ .

$25 {(x^{\frac{1}{2}})}^{2} = {(- x - 6)}^{2}$

Step 3.2.1.3

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

Tap for more steps...

Step 3.2.1.3.1

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

$25 x^{\frac{1}{2} \cdot 2} = {(- x - 6)}^{2}$

Step 3.2.1.3.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.2.1.3.2.1

Cancel the common factor.

$25 x^{\frac{1}{2} \cdot 2} = {(- x - 6)}^{2}$

Step 3.2.1.3.2.2

Rewrite the expression.

$25 x^{1} = {(- x - 6)}^{2}$

Step 3.2.1.4

Simplify.

$25 x = {(- x - 6)}^{2}$

Step 3.3

Simplify the right side.

Tap for more steps...

Step 3.3.1

Simplify ${(- x - 6)}^{2}$ .

Tap for more steps...

Step 3.3.1.1

Rewrite ${(- x - 6)}^{2}$ as $(- x - 6) (- x - 6)$ .

$25 x = (- x - 6) (- x - 6)$

Step 3.3.1.2

Expand $(- x - 6) (- x - 6)$ using the FOIL Method.

Tap for more steps...

Step 3.3.1.2.1

Apply the distributive property.

$25 x = - x (- x - 6) - 6 (- x - 6)$

Step 3.3.1.2.2

Apply the distributive property.

$25 x = - x (- x) - x \cdot - 6 - 6 (- x - 6)$

Step 3.3.1.2.3

Apply the distributive property.

$25 x = - x (- x) - x \cdot - 6 - 6 (- x) - 6 \cdot - 6$

Step 3.3.1.3

Simplify and combine like terms.

Tap for more steps...

Step 3.3.1.3.1

Simplify each term.

Tap for more steps...

Step 3.3.1.3.1.1

Rewrite using the commutative property of multiplication.

$25 x = - 1 \cdot - 1 x \cdot x - x \cdot - 6 - 6 (- x) - 6 \cdot - 6$

Step 3.3.1.3.1.2

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 3.3.1.3.1.2.1

Move $x$ .

$25 x = - 1 \cdot - 1 (x \cdot x) - x \cdot - 6 - 6 (- x) - 6 \cdot - 6$

Step 3.3.1.3.1.2.2

Multiply $x$ by $x$ .

$25 x = - 1 \cdot - 1 x^{2} - x \cdot - 6 - 6 (- x) - 6 \cdot - 6$

Step 3.3.1.3.1.3

Multiply $- 1$ by $- 1$ .

$25 x = 1 x^{2} - x \cdot - 6 - 6 (- x) - 6 \cdot - 6$

Step 3.3.1.3.1.4

Multiply $x^{2}$ by $1$ .

$25 x = x^{2} - x \cdot - 6 - 6 (- x) - 6 \cdot - 6$

Step 3.3.1.3.1.5

Multiply $- 6$ by $- 1$ .

$25 x = x^{2} + 6 x - 6 (- x) - 6 \cdot - 6$

Step 3.3.1.3.1.6

Multiply $- 1$ by $- 6$ .

$25 x = x^{2} + 6 x + 6 x - 6 \cdot - 6$

Step 3.3.1.3.1.7

Multiply $- 6$ by $- 6$ .

$25 x = x^{2} + 6 x + 6 x + 36$

Step 3.3.1.3.2

Add $6 x$ and $6 x$ .

$25 x = x^{2} + 12 x + 36$

Step 4

Solve for $x$ .

Tap for more steps...

Step 4.1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$x^{2} + 12 x + 36 = 25 x$

Step 4.2

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

Tap for more steps...

Step 4.2.1

Subtract $25 x$ from both sides of the equation.

$x^{2} + 12 x + 36 - 25 x = 0$

Step 4.2.2

Subtract $25 x$ from $12 x$ .

$x^{2} - 13 x + 36 = 0$

Step 4.3

Factor $x^{2} - 13 x + 36$ using the AC method.

Tap for more steps...

Step 4.3.1

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 $36$ and whose sum is $- 13$ .

$- 9, - 4$

Step 4.3.2

Write the factored form using these integers.

$(x - 9) (x - 4) = 0$

Step 4.4

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

$x - 9 = 0$

$x - 4 = 0$

Step 4.5

Set $x - 9$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 4.5.1

Set $x - 9$ equal to $0$ .

$x - 9 = 0$

Step 4.5.2

Add $9$ to both sides of the equation.

$x = 9$

Step 4.6

Set $x - 4$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 4.6.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 4.6.2

Add $4$ to both sides of the equation.

$x = 4$

Step 4.7

The final solution is all the values that make $(x - 9) (x - 4) = 0$ true.

$x = 9, 4$