Pre-Algebra Examples

$\frac{1}{2} \cdot (x (5 x - 45)) = 45$

Step 1

Multiply both sides of the equation by $2$ .

$2 (\frac{1}{2} \cdot (x (5 x - 45))) = 2 \cdot 45$

Step 2

Simplify both sides of the equation.

Tap for more steps...

Step 2.1

Simplify the left side.

Tap for more steps...

Step 2.1.1

Simplify $2 (\frac{1}{2} \cdot (x (5 x - 45)))$ .

Tap for more steps...

Step 2.1.1.1

Simplify by multiplying through.

Tap for more steps...

Step 2.1.1.1.1

Apply the distributive property.

$2 (\frac{1}{2} \cdot (x (5 x) + x \cdot - 45)) = 2 \cdot 45$

Step 2.1.1.1.2

Reorder.

Tap for more steps...

Step 2.1.1.1.2.1

Rewrite using the commutative property of multiplication.

$2 (\frac{1}{2} \cdot (5 x \cdot x + x \cdot - 45)) = 2 \cdot 45$

Step 2.1.1.1.2.2

Move $- 45$ to the left of $x$ .

$2 (\frac{1}{2} \cdot (5 x \cdot x - 45 \cdot x)) = 2 \cdot 45$

Step 2.1.1.2

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

Tap for more steps...

Step 2.1.1.2.1

Move $x$ .

$2 (\frac{1}{2} \cdot (5 (x \cdot x) - 45 \cdot x)) = 2 \cdot 45$

Step 2.1.1.2.2

Multiply $x$ by $x$ .

$2 (\frac{1}{2} \cdot (5 x^{2} - 45 \cdot x)) = 2 \cdot 45$

$2 (\frac{1}{2} \cdot (5 x^{2} - 45 x)) = 2 \cdot 45$

Step 2.1.1.3

Apply the distributive property.

$2 (\frac{1}{2} (5 x^{2}) + \frac{1}{2} (- 45 x)) = 2 \cdot 45$

Step 2.1.1.4

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

Tap for more steps...

Step 2.1.1.4.1

Combine $5$ and $\frac{1}{2}$ .

$2 (\frac{5}{2} x^{2} + \frac{1}{2} (- 45 x)) = 2 \cdot 45$

Step 2.1.1.4.2

Combine $\frac{5}{2}$ and $x^{2}$ .

$2 (\frac{5 x^{2}}{2} + \frac{1}{2} (- 45 x)) = 2 \cdot 45$

Step 2.1.1.5

Multiply $\frac{1}{2} (- 45 x)$ .

Tap for more steps...

Step 2.1.1.5.1

Combine $- 45$ and $\frac{1}{2}$ .

$2 (\frac{5 x^{2}}{2} + \frac{- 45}{2} x) = 2 \cdot 45$

Step 2.1.1.5.2

Combine $\frac{- 45}{2}$ and $x$ .

$2 (\frac{5 x^{2}}{2} + \frac{- 45 x}{2}) = 2 \cdot 45$

Step 2.1.1.6

Move the negative in front of the fraction.

$2 (\frac{5 x^{2}}{2} - \frac{45 x}{2}) = 2 \cdot 45$

Step 2.1.1.7

Apply the distributive property.

$2 \frac{5 x^{2}}{2} + 2 (- \frac{45 x}{2}) = 2 \cdot 45$

Step 2.1.1.8

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.1.1.8.1

Cancel the common factor.

$2 \frac{5 x^{2}}{2} + 2 (- \frac{45 x}{2}) = 2 \cdot 45$

Step 2.1.1.8.2

Rewrite the expression.

$5 x^{2} + 2 (- \frac{45 x}{2}) = 2 \cdot 45$

Step 2.1.1.9

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.1.1.9.1

Move the leading negative in $- \frac{45 x}{2}$ into the numerator.

$5 x^{2} + 2 \frac{- 45 x}{2} = 2 \cdot 45$

Step 2.1.1.9.2

Cancel the common factor.

$5 x^{2} + 2 \frac{- 45 x}{2} = 2 \cdot 45$

Step 2.1.1.9.3

Rewrite the expression.

$5 x^{2} - 45 x = 2 \cdot 45$

Step 2.2

Simplify the right side.

Tap for more steps...

Step 2.2.1

Multiply $2$ by $45$ .

$5 x^{2} - 45 x = 90$

Step 3

Subtract $90$ from both sides of the equation.

$5 x^{2} - 45 x - 90 = 0$

Step 4

Factor $5$ out of $5 x^{2} - 45 x - 90$ .

Tap for more steps...

Step 4.1

Factor $5$ out of $5 x^{2}$ .

$5 (x^{2}) - 45 x - 90 = 0$

Step 4.2

Factor $5$ out of $- 45 x$ .

$5 (x^{2}) + 5 (- 9 x) - 90 = 0$

Step 4.3

Factor $5$ out of $- 90$ .

$5 x^{2} + 5 (- 9 x) + 5 \cdot - 18 = 0$

Step 4.4

Factor $5$ out of $5 x^{2} + 5 (- 9 x)$ .

$5 (x^{2} - 9 x) + 5 \cdot - 18 = 0$

Step 4.5

Factor $5$ out of $5 (x^{2} - 9 x) + 5 \cdot - 18$ .

$5 (x^{2} - 9 x - 18) = 0$

Step 5

Divide each term in $5 (x^{2} - 9 x - 18) = 0$ by $5$ and simplify.

Tap for more steps...

Step 5.1

Divide each term in $5 (x^{2} - 9 x - 18) = 0$ by $5$ .

$\frac{5 (x^{2} - 9 x - 18)}{5} = \frac{0}{5}$

Step 5.2

Simplify the left side.

Tap for more steps...

Step 5.2.1

Cancel the common factor of $5$ .

Tap for more steps...

Step 5.2.1.1

Cancel the common factor.

$\frac{5 (x^{2} - 9 x - 18)}{5} = \frac{0}{5}$

Step 5.2.1.2

Divide $x^{2} - 9 x - 18$ by $1$ .

$x^{2} - 9 x - 18 = \frac{0}{5}$

Step 5.3

Simplify the right side.

Tap for more steps...

Step 5.3.1

Divide $0$ by $5$ .

$x^{2} - 9 x - 18 = 0$

Step 6

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 7

Substitute the values $a = 1$ , $b = - 9$ , and $c = - 18$ into the quadratic formula and solve for $x$ .

$\frac{9 \pm \sqrt{{(- 9)}^{2} - 4 \cdot (1 \cdot - 18)}}{2 \cdot 1}$

Step 8

Simplify.

Tap for more steps...

Step 8.1

Simplify the numerator.

Tap for more steps...

Step 8.1.1

Raise $- 9$ to the power of $2$ .

$x = \frac{9 \pm \sqrt{81 - 4 \cdot 1 \cdot - 18}}{2 \cdot 1}$

Step 8.1.2

Multiply $- 4 \cdot 1 \cdot - 18$ .

Tap for more steps...

Step 8.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{9 \pm \sqrt{81 - 4 \cdot - 18}}{2 \cdot 1}$

Step 8.1.2.2

Multiply $- 4$ by $- 18$ .

$x = \frac{9 \pm \sqrt{81 + 72}}{2 \cdot 1}$

Step 8.1.3

Add $81$ and $72$ .

$x = \frac{9 \pm \sqrt{153}}{2 \cdot 1}$

Step 8.1.4

Rewrite $153$ as $3^{2} \cdot 17$ .

Tap for more steps...

Step 8.1.4.1

Factor $9$ out of $153$ .

$x = \frac{9 \pm \sqrt{9 (17)}}{2 \cdot 1}$

Step 8.1.4.2

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

$x = \frac{9 \pm \sqrt{3^{2} \cdot 17}}{2 \cdot 1}$

Step 8.1.5

Pull terms out from under the radical.

$x = \frac{9 \pm 3 \sqrt{17}}{2 \cdot 1}$

Step 8.2

Multiply $2$ by $1$ .

$x = \frac{9 \pm 3 \sqrt{17}}{2}$

Step 9

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Tap for more steps...

Step 9.1

Simplify the numerator.

Tap for more steps...

Step 9.1.1

Raise $- 9$ to the power of $2$ .

$x = \frac{9 \pm \sqrt{81 - 4 \cdot 1 \cdot - 18}}{2 \cdot 1}$

Step 9.1.2

Multiply $- 4 \cdot 1 \cdot - 18$ .

Tap for more steps...

Step 9.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{9 \pm \sqrt{81 - 4 \cdot - 18}}{2 \cdot 1}$

Step 9.1.2.2

Multiply $- 4$ by $- 18$ .

$x = \frac{9 \pm \sqrt{81 + 72}}{2 \cdot 1}$

Step 9.1.3

Add $81$ and $72$ .

$x = \frac{9 \pm \sqrt{153}}{2 \cdot 1}$

Step 9.1.4

Rewrite $153$ as $3^{2} \cdot 17$ .

Tap for more steps...

Step 9.1.4.1

Factor $9$ out of $153$ .

$x = \frac{9 \pm \sqrt{9 (17)}}{2 \cdot 1}$

Step 9.1.4.2

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

$x = \frac{9 \pm \sqrt{3^{2} \cdot 17}}{2 \cdot 1}$

Step 9.1.5

Pull terms out from under the radical.

$x = \frac{9 \pm 3 \sqrt{17}}{2 \cdot 1}$

Step 9.2

Multiply $2$ by $1$ .

$x = \frac{9 \pm 3 \sqrt{17}}{2}$

Step 9.3

Change the $\pm$ to $+$ .

$x = \frac{9 + 3 \sqrt{17}}{2}$

Step 10

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Tap for more steps...

Step 10.1

Simplify the numerator.

Tap for more steps...

Step 10.1.1

Raise $- 9$ to the power of $2$ .

$x = \frac{9 \pm \sqrt{81 - 4 \cdot 1 \cdot - 18}}{2 \cdot 1}$

Step 10.1.2

Multiply $- 4 \cdot 1 \cdot - 18$ .

Tap for more steps...

Step 10.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{9 \pm \sqrt{81 - 4 \cdot - 18}}{2 \cdot 1}$

Step 10.1.2.2

Multiply $- 4$ by $- 18$ .

$x = \frac{9 \pm \sqrt{81 + 72}}{2 \cdot 1}$

Step 10.1.3

Add $81$ and $72$ .

$x = \frac{9 \pm \sqrt{153}}{2 \cdot 1}$

Step 10.1.4

Rewrite $153$ as $3^{2} \cdot 17$ .

Tap for more steps...

Step 10.1.4.1

Factor $9$ out of $153$ .

$x = \frac{9 \pm \sqrt{9 (17)}}{2 \cdot 1}$

Step 10.1.4.2

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

$x = \frac{9 \pm \sqrt{3^{2} \cdot 17}}{2 \cdot 1}$

Step 10.1.5

Pull terms out from under the radical.

$x = \frac{9 \pm 3 \sqrt{17}}{2 \cdot 1}$

Step 10.2

Multiply $2$ by $1$ .

$x = \frac{9 \pm 3 \sqrt{17}}{2}$

Step 10.3

Change the $\pm$ to $-$ .

$x = \frac{9 - 3 \sqrt{17}}{2}$

Step 11

The final answer is the combination of both solutions.

$x = \frac{9 + 3 \sqrt{17}}{2}, \frac{9 - 3 \sqrt{17}}{2}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$x = \frac{9 + 3 \sqrt{17}}{2}, \frac{9 - 3 \sqrt{17}}{2}$

Decimal Form:

$x = 10.68465843 \dots, - 1.68465843 \dots$