Finite Math Examples

$x (11 x - 2) = 5$

Step 1

Move all terms to the left side of the equation and simplify.

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Step 1.1

Simplify the left side.

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Step 1.1.1

Simplify $x (11 x - 2)$ .

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Step 1.1.1.1

Simplify by multiplying through.

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Step 1.1.1.1.1

Apply the distributive property.

$x (11 x) + x \cdot - 2 = 5$

Step 1.1.1.1.2

Reorder.

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Step 1.1.1.1.2.1

Rewrite using the commutative property of multiplication.

$11 x \cdot x + x \cdot - 2 = 5$

Step 1.1.1.1.2.2

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

$11 x \cdot x - 2 \cdot x = 5$

Step 1.1.1.2

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

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Step 1.1.1.2.1

Move $x$ .

$11 (x \cdot x) - 2 \cdot x = 5$

Step 1.1.1.2.2

Multiply $x$ by $x$ .

$11 x^{2} - 2 \cdot x = 5$

$11 x^{2} - 2 x = 5$

Step 1.2

Subtract $5$ from both sides of the equation.

$11 x^{2} - 2 x - 5 = 0$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values $a = 11$ , $b = - 2$ , and $c = - 5$ into the quadratic formula and solve for $x$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (11 \cdot - 5)}}{2 \cdot 11}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 11 \cdot - 5}}{2 \cdot 11}$

Step 4.1.2

Multiply $- 4 \cdot 11 \cdot - 5$ .

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Step 4.1.2.1

Multiply $- 4$ by $11$ .

$x = \frac{2 \pm \sqrt{4 - 44 \cdot - 5}}{2 \cdot 11}$

Step 4.1.2.2

Multiply $- 44$ by $- 5$ .

$x = \frac{2 \pm \sqrt{4 + 220}}{2 \cdot 11}$

Step 4.1.3

Add $4$ and $220$ .

$x = \frac{2 \pm \sqrt{224}}{2 \cdot 11}$

Step 4.1.4

Rewrite $224$ as $4^{2} \cdot 14$ .

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Step 4.1.4.1

Factor $16$ out of $224$ .

$x = \frac{2 \pm \sqrt{16 (14)}}{2 \cdot 11}$

Step 4.1.4.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{2 \pm \sqrt{4^{2} \cdot 14}}{2 \cdot 11}$

Step 4.1.5

Pull terms out from under the radical.

$x = \frac{2 \pm 4 \sqrt{14}}{2 \cdot 11}$

Step 4.2

Multiply $2$ by $11$ .

$x = \frac{2 \pm 4 \sqrt{14}}{22}$

Step 4.3

Simplify $\frac{2 \pm 4 \sqrt{14}}{22}$ .

$x = \frac{1 \pm 2 \sqrt{14}}{11}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1 \pm 2 \sqrt{14}}{11}$

Decimal Form:

$x = 0.77121043 \dots, - 0.58939225 \dots$