Pre-Algebra Examples

$0.6 x^{2} = 1.2 x + 0.6$

Step 1

Subtract $1.2 x$ from both sides of the equation.

$0.6 x^{2} - 1.2 x = 0.6$

Step 2

Subtract $0.6$ from both sides of the equation.

$0.6 x^{2} - 1.2 x - 0.6 = 0$

Step 3

Factor $0.6$ out of $0.6 x^{2} - 1.2 x - 0.6$ .

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Step 3.1

Factor $0.6$ out of $0.6 x^{2}$ .

$0.6 (x^{2}) - 1.2 x - 0.6 = 0$

Step 3.2

Factor $0.6$ out of $- 1.2 x$ .

$0.6 (x^{2}) + 0.6 (- 2 x) - 0.6 = 0$

Step 3.3

Factor $0.6$ out of $- 0.6$ .

$0.6 (x^{2}) + 0.6 (- 2 x) + 0.6 (- 1) = 0$

Step 3.4

Factor $0.6$ out of $0.6 (x^{2}) + 0.6 (- 2 x)$ .

$0.6 (x^{2} - 2 x) + 0.6 (- 1) = 0$

Step 3.5

Factor $0.6$ out of $0.6 (x^{2} - 2 x) + 0.6 (- 1)$ .

$0.6 (x^{2} - 2 x - 1) = 0$

Step 4

Divide each term in $0.6 (x^{2} - 2 x - 1) = 0$ by $0.6$ and simplify.

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Step 4.1

Divide each term in $0.6 (x^{2} - 2 x - 1) = 0$ by $0.6$ .

$\frac{0.6 (x^{2} - 2 x - 1)}{0.6} = \frac{0}{0.6}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $0.6$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{0.6 (x^{2} - 2 x - 1)}{0.6} = \frac{0}{0.6}$

Step 4.2.1.2

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

$x^{2} - 2 x - 1 = \frac{0}{0.6}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $0$ by $0.6$ .

$x^{2} - 2 x - 1 = 0$

Step 5

Use the quadratic formula to find the solutions.

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

Step 6

Substitute the values $a = 1$ , $b = - 2$ , and $c = - 1$ into the quadratic formula and solve for $x$ .

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

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

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

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

Step 7.1.2

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

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Step 7.1.2.1

Multiply $- 4$ by $1$ .

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

Step 7.1.2.2

Multiply $- 4$ by $- 1$ .

$x = \frac{2 \pm \sqrt{4 + 4}}{2 \cdot 1}$

Step 7.1.3

Add $4$ and $4$ .

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

Step 7.1.4

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

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Step 7.1.4.1

Factor $4$ out of $8$ .

$x = \frac{2 \pm \sqrt{4 (2)}}{2 \cdot 1}$

Step 7.1.4.2

Rewrite $4$ as $2^{2}$ .

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

Step 7.1.5

Pull terms out from under the radical.

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

Step 7.2

Multiply $2$ by $1$ .

$x = \frac{2 \pm 2 \sqrt{2}}{2}$

Step 7.3

Simplify $\frac{2 \pm 2 \sqrt{2}}{2}$ .

$x = 1 \pm \sqrt{2}$

Step 8

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

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Step 8.1

Simplify the numerator.

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Step 8.1.1

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

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

Step 8.1.2

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

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Step 8.1.2.1

Multiply $- 4$ by $1$ .

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

Step 8.1.2.2

Multiply $- 4$ by $- 1$ .

$x = \frac{2 \pm \sqrt{4 + 4}}{2 \cdot 1}$

Step 8.1.3

Add $4$ and $4$ .

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

Step 8.1.4

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

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Step 8.1.4.1

Factor $4$ out of $8$ .

$x = \frac{2 \pm \sqrt{4 (2)}}{2 \cdot 1}$

Step 8.1.4.2

Rewrite $4$ as $2^{2}$ .

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

Step 8.1.5

Pull terms out from under the radical.

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

Step 8.2

Multiply $2$ by $1$ .

$x = \frac{2 \pm 2 \sqrt{2}}{2}$

Step 8.3

Simplify $\frac{2 \pm 2 \sqrt{2}}{2}$ .

$x = 1 \pm \sqrt{2}$

Step 8.4

Change the $\pm$ to $+$ .

$x = 1 + \sqrt{2}$

Step 9

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

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Step 9.1

Simplify the numerator.

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Step 9.1.1

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

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

Step 9.1.2

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

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Step 9.1.2.1

Multiply $- 4$ by $1$ .

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

Step 9.1.2.2

Multiply $- 4$ by $- 1$ .

$x = \frac{2 \pm \sqrt{4 + 4}}{2 \cdot 1}$

Step 9.1.3

Add $4$ and $4$ .

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

Step 9.1.4

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

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Step 9.1.4.1

Factor $4$ out of $8$ .

$x = \frac{2 \pm \sqrt{4 (2)}}{2 \cdot 1}$

Step 9.1.4.2

Rewrite $4$ as $2^{2}$ .

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

Step 9.1.5

Pull terms out from under the radical.

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

Step 9.2

Multiply $2$ by $1$ .

$x = \frac{2 \pm 2 \sqrt{2}}{2}$

Step 9.3

Simplify $\frac{2 \pm 2 \sqrt{2}}{2}$ .

$x = 1 \pm \sqrt{2}$

Step 9.4

Change the $\pm$ to $-$ .

$x = 1 - \sqrt{2}$

Step 10

The final answer is the combination of both solutions.

$x = 1 + \sqrt{2}, 1 - \sqrt{2}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$x = 1 + \sqrt{2}, 1 - \sqrt{2}$

Decimal Form:

$x = 2.41421356 \dots, - 0.41421356 \dots$