Algebra Examples

$x \cdot 0.5 (x - 10) = 30 - 15 x$

Step 1

Simplify $x \cdot (0.5 (x - 10))$ .

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Step 1.1

Rewrite.

$0 + 0 + x \cdot (0.5 (x - 10)) = 30 - 15 x$

Step 1.2

Rewrite using the commutative property of multiplication.

$0.5 x (x - 10) = 30 - 15 x$

Step 1.3

Apply the distributive property.

$0.5 x \cdot x + 0.5 x \cdot - 10 = 30 - 15 x$

Step 1.4

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

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Step 1.4.1

Move $x$ .

$0.5 (x \cdot x) + 0.5 x \cdot - 10 = 30 - 15 x$

Step 1.4.2

Multiply $x$ by $x$ .

$0.5 x^{2} + 0.5 x \cdot - 10 = 30 - 15 x$

Step 1.5

Multiply $- 10$ by $0.5$ .

$0.5 x^{2} - 5 x = 30 - 15 x$

Step 2

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

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Step 2.1

Add $15 x$ to both sides of the equation.

$0.5 x^{2} - 5 x + 15 x = 30$

Step 2.2

Add $- 5 x$ and $15 x$ .

$0.5 x^{2} + 10 x = 30$

Step 3

Subtract $30$ from both sides of the equation.

$0.5 x^{2} + 10 x - 30 = 0$

Step 4

Factor $0.5$ out of $0.5 x^{2} + 10 x - 30$ .

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Step 4.1

Factor $0.5$ out of $0.5 x^{2}$ .

$0.5 (x^{2}) + 10 x - 30 = 0$

Step 4.2

Factor $0.5$ out of $10 x$ .

$0.5 (x^{2}) + 0.5 (20 x) - 30 = 0$

Step 4.3

Factor $0.5$ out of $- 30$ .

$0.5 x^{2} + 0.5 (20 x) + 0.5 \cdot - 60 = 0$

Step 4.4

Factor $0.5$ out of $0.5 x^{2} + 0.5 (20 x)$ .

$0.5 (x^{2} + 20 x) + 0.5 \cdot - 60 = 0$

Step 4.5

Factor $0.5$ out of $0.5 (x^{2} + 20 x) + 0.5 \cdot - 60$ .

$0.5 (x^{2} + 20 x - 60) = 0$

Step 5

Divide each term in $0.5 (x^{2} + 20 x - 60) = 0$ by $0.5$ and simplify.

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Step 5.1

Divide each term in $0.5 (x^{2} + 20 x - 60) = 0$ by $0.5$ .

$\frac{0.5 (x^{2} + 20 x - 60)}{0.5} = \frac{0}{0.5}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $0.5$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{0.5 (x^{2} + 20 x - 60)}{0.5} = \frac{0}{0.5}$

Step 5.2.1.2

Divide $x^{2} + 20 x - 60$ by $1$ .

$x^{2} + 20 x - 60 = \frac{0}{0.5}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Divide $0$ by $0.5$ .

$x^{2} + 20 x - 60 = 0$

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 $a = 1$ , $b = 20$ , and $c = - 60$ into the quadratic formula and solve for $x$ .

$\frac{- 20 \pm \sqrt{20^{2} - 4 \cdot (1 \cdot - 60)}}{2 \cdot 1}$

Step 8

Simplify.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raise $20$ to the power of $2$ .

$x = \frac{- 20 \pm \sqrt{400 - 4 \cdot 1 \cdot - 60}}{2 \cdot 1}$

Step 8.1.2

Multiply $- 4 \cdot 1 \cdot - 60$ .

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Step 8.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 20 \pm \sqrt{400 - 4 \cdot - 60}}{2 \cdot 1}$

Step 8.1.2.2

Multiply $- 4$ by $- 60$ .

$x = \frac{- 20 \pm \sqrt{400 + 240}}{2 \cdot 1}$

Step 8.1.3

Add $400$ and $240$ .

$x = \frac{- 20 \pm \sqrt{640}}{2 \cdot 1}$

Step 8.1.4

Rewrite $640$ as $8^{2} \cdot 10$ .

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Step 8.1.4.1

Factor $64$ out of $640$ .

$x = \frac{- 20 \pm \sqrt{64 (10)}}{2 \cdot 1}$

Step 8.1.4.2

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

$x = \frac{- 20 \pm \sqrt{8^{2} \cdot 10}}{2 \cdot 1}$

Step 8.1.5

Pull terms out from under the radical.

$x = \frac{- 20 \pm 8 \sqrt{10}}{2 \cdot 1}$

Step 8.2

Multiply $2$ by $1$ .

$x = \frac{- 20 \pm 8 \sqrt{10}}{2}$

Step 8.3

Simplify $\frac{- 20 \pm 8 \sqrt{10}}{2}$ .

$x = - 10 \pm 4 \sqrt{10}$

Step 9

The final answer is the combination of both solutions.

$x = - 10 + 4 \sqrt{10}, - 10 - 4 \sqrt{10}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = - 10 + 4 \sqrt{10}, - 10 - 4 \sqrt{10}$

Decimal Form:

$x = 2.64911064 \dots, - 22.64911064 \dots$