Pre-Algebra Examples

$x (10 x - 2) = 11$

Step 1

Simplify $x (10 x - 2)$ .

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Step 1.1

Simplify by multiplying through.

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Step 1.1.1

Apply the distributive property.

$x (10 x) + x \cdot - 2 = 11$

Step 1.1.2

Reorder.

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Step 1.1.2.1

Rewrite using the commutative property of multiplication.

$10 x \cdot x + x \cdot - 2 = 11$

Step 1.1.2.2

Move $- 2$ to the left of $x$ .

$10 x \cdot x - 2 \cdot x = 11$

Step 1.2

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

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Step 1.2.1

Move $x$ .

$10 (x \cdot x) - 2 \cdot x = 11$

Step 1.2.2

Multiply $x$ by $x$ .

$10 x^{2} - 2 \cdot x = 11$

$10 x^{2} - 2 x = 11$

Step 2

Subtract $11$ from both sides of the equation.

$10 x^{2} - 2 x - 11 = 0$

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = 10$ , $b = - 2$ , and $c = - 11$ into the quadratic formula and solve for $x$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (10 \cdot - 11)}}{2 \cdot 10}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 10 \cdot - 11}}{2 \cdot 10}$

Step 5.1.2

Multiply $- 4 \cdot 10 \cdot - 11$ .

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Step 5.1.2.1

Multiply $- 4$ by $10$ .

$x = \frac{2 \pm \sqrt{4 - 40 \cdot - 11}}{2 \cdot 10}$

Step 5.1.2.2

Multiply $- 40$ by $- 11$ .

$x = \frac{2 \pm \sqrt{4 + 440}}{2 \cdot 10}$

Step 5.1.3

Add $4$ and $440$ .

$x = \frac{2 \pm \sqrt{444}}{2 \cdot 10}$

Step 5.1.4

Rewrite $444$ as $2^{2} \cdot 111$ .

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Step 5.1.4.1

Factor $4$ out of $444$ .

$x = \frac{2 \pm \sqrt{4 (111)}}{2 \cdot 10}$

Step 5.1.4.2

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

$x = \frac{2 \pm \sqrt{2^{2} \cdot 111}}{2 \cdot 10}$

Step 5.1.5

Pull terms out from under the radical.

$x = \frac{2 \pm 2 \sqrt{111}}{2 \cdot 10}$

Step 5.2

Multiply $2$ by $10$ .

$x = \frac{2 \pm 2 \sqrt{111}}{20}$

Step 5.3

Simplify $\frac{2 \pm 2 \sqrt{111}}{20}$ .

$x = \frac{1 \pm \sqrt{111}}{10}$

Step 6

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

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Step 6.1

Simplify the numerator.

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Step 6.1.1

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 10 \cdot - 11}}{2 \cdot 10}$

Step 6.1.2

Multiply $- 4 \cdot 10 \cdot - 11$ .

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Step 6.1.2.1

Multiply $- 4$ by $10$ .

$x = \frac{2 \pm \sqrt{4 - 40 \cdot - 11}}{2 \cdot 10}$

Step 6.1.2.2

Multiply $- 40$ by $- 11$ .

$x = \frac{2 \pm \sqrt{4 + 440}}{2 \cdot 10}$

Step 6.1.3

Add $4$ and $440$ .

$x = \frac{2 \pm \sqrt{444}}{2 \cdot 10}$

Step 6.1.4

Rewrite $444$ as $2^{2} \cdot 111$ .

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Step 6.1.4.1

Factor $4$ out of $444$ .

$x = \frac{2 \pm \sqrt{4 (111)}}{2 \cdot 10}$

Step 6.1.4.2

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

$x = \frac{2 \pm \sqrt{2^{2} \cdot 111}}{2 \cdot 10}$

Step 6.1.5

Pull terms out from under the radical.

$x = \frac{2 \pm 2 \sqrt{111}}{2 \cdot 10}$

Step 6.2

Multiply $2$ by $10$ .

$x = \frac{2 \pm 2 \sqrt{111}}{20}$

Step 6.3

Simplify $\frac{2 \pm 2 \sqrt{111}}{20}$ .

$x = \frac{1 \pm \sqrt{111}}{10}$

Step 6.4

Change the $\pm$ to $+$ .

$x = \frac{1 + \sqrt{111}}{10}$

Step 7

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

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Step 7.1

Simplify the numerator.

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Step 7.1.1

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 10 \cdot - 11}}{2 \cdot 10}$

Step 7.1.2

Multiply $- 4 \cdot 10 \cdot - 11$ .

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Step 7.1.2.1

Multiply $- 4$ by $10$ .

$x = \frac{2 \pm \sqrt{4 - 40 \cdot - 11}}{2 \cdot 10}$

Step 7.1.2.2

Multiply $- 40$ by $- 11$ .

$x = \frac{2 \pm \sqrt{4 + 440}}{2 \cdot 10}$

Step 7.1.3

Add $4$ and $440$ .

$x = \frac{2 \pm \sqrt{444}}{2 \cdot 10}$

Step 7.1.4

Rewrite $444$ as $2^{2} \cdot 111$ .

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Step 7.1.4.1

Factor $4$ out of $444$ .

$x = \frac{2 \pm \sqrt{4 (111)}}{2 \cdot 10}$

Step 7.1.4.2

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

$x = \frac{2 \pm \sqrt{2^{2} \cdot 111}}{2 \cdot 10}$

Step 7.1.5

Pull terms out from under the radical.

$x = \frac{2 \pm 2 \sqrt{111}}{2 \cdot 10}$

Step 7.2

Multiply $2$ by $10$ .

$x = \frac{2 \pm 2 \sqrt{111}}{20}$

Step 7.3

Simplify $\frac{2 \pm 2 \sqrt{111}}{20}$ .

$x = \frac{1 \pm \sqrt{111}}{10}$

Step 7.4

Change the $\pm$ to $-$ .

$x = \frac{1 - \sqrt{111}}{10}$

Step 8

The final answer is the combination of both solutions.

$x = \frac{1 + \sqrt{111}}{10}, \frac{1 - \sqrt{111}}{10}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1 + \sqrt{111}}{10}, \frac{1 - \sqrt{111}}{10}$

Decimal Form:

$x = 1.15356537 \dots, - 0.95356537 \dots$