Finite Math Examples

$- 0.7 x (x - 11) = 1.8 x + 5.4$

Step 1

Simplify $- 0.7 x (x - 11)$ .

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Step 1.1

Rewrite.

$0 + 0 - 0.7 x (x - 11) = 1.8 x + 5.4$

Step 1.2

Simplify by adding zeros.

$- 0.7 x (x - 11) = 1.8 x + 5.4$

Step 1.3

Apply the distributive property.

$- 0.7 x \cdot x - 0.7 x \cdot - 11 = 1.8 x + 5.4$

Step 1.4

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

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Step 1.4.1

Move $x$ .

$- 0.7 (x \cdot x) - 0.7 x \cdot - 11 = 1.8 x + 5.4$

Step 1.4.2

Multiply $x$ by $x$ .

$- 0.7 x^{2} - 0.7 x \cdot - 11 = 1.8 x + 5.4$

Step 1.5

Multiply $- 11$ by $- 0.7$ .

$- 0.7 x^{2} + 7.7 x = 1.8 x + 5.4$

Step 2

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

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Step 2.1

Subtract $1.8 x$ from both sides of the equation.

$- 0.7 x^{2} + 7.7 x - 1.8 x = 5.4$

Step 2.2

Subtract $1.8 x$ from $7.7 x$ .

$- 0.7 x^{2} + 5.9 x = 5.4$

Step 3

Subtract $5.4$ from both sides of the equation.

$- 0.7 x^{2} + 5.9 x - 5.4 = 0$

Step 4

Factor $- 0.1$ out of $- 0.7 x^{2} + 5.9 x - 5.4$ .

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Step 4.1

Factor $- 0.1$ out of $- 0.7 x^{2}$ .

$- 0.1 (7 x^{2}) + 5.9 x - 5.4 = 0$

Step 4.2

Factor $- 0.1$ out of $5.9 x$ .

$- 0.1 (7 x^{2}) - 0.1 (- 59 x) - 5.4 = 0$

Step 4.3

Factor $- 0.1$ out of $- 5.4$ .

$- 0.1 (7 x^{2}) - 0.1 (- 59 x) - 0.1 \cdot 54 = 0$

Step 4.4

Factor $- 0.1$ out of $- 0.1 (7 x^{2}) - 0.1 (- 59 x)$ .

$- 0.1 (7 x^{2} - 59 x) - 0.1 \cdot 54 = 0$

Step 4.5

Factor $- 0.1$ out of $- 0.1 (7 x^{2} - 59 x) - 0.1 (54)$ .

$- 0.1 (7 x^{2} - 59 x + 54) = 0$

Step 5

Divide each term in $- 0.1 (7 x^{2} - 59 x + 54) = 0$ by $- 0.1$ and simplify.

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Step 5.1

Divide each term in $- 0.1 (7 x^{2} - 59 x + 54) = 0$ by $- 0.1$ .

$\frac{- 0.1 (7 x^{2} - 59 x + 54)}{- 0.1} = \frac{0}{- 0.1}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $- 0.1$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{- 0.1 (7 x^{2} - 59 x + 54)}{- 0.1} = \frac{0}{- 0.1}$

Step 5.2.1.2

Divide $7 x^{2} - 59 x + 54$ by $1$ .

$7 x^{2} - 59 x + 54 = \frac{0}{- 0.1}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Divide $0$ by $- 0.1$ .

$7 x^{2} - 59 x + 54 = 0$

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 $a = 7$ , $b = - 59$ , and $c = 54$ into the quadratic formula and solve for $x$ .

$\frac{59 \pm \sqrt{{(- 59)}^{2} - 4 \cdot (7 \cdot 54)}}{2 \cdot 7}$

Step 8

Simplify.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raise $- 59$ to the power of $2$ .

$x = \frac{59 \pm \sqrt{3481 - 4 \cdot 7 \cdot 54}}{2 \cdot 7}$

Step 8.1.2

Multiply $- 4 \cdot 7 \cdot 54$ .

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Step 8.1.2.1

Multiply $- 4$ by $7$ .

$x = \frac{59 \pm \sqrt{3481 - 28 \cdot 54}}{2 \cdot 7}$

Step 8.1.2.2

Multiply $- 28$ by $54$ .

$x = \frac{59 \pm \sqrt{3481 - 1512}}{2 \cdot 7}$

Step 8.1.3

Subtract $1512$ from $3481$ .

$x = \frac{59 \pm \sqrt{1969}}{2 \cdot 7}$

Step 8.2

Multiply $2$ by $7$ .

$x = \frac{59 \pm \sqrt{1969}}{14}$

Step 9

The final answer is the combination of both solutions.

$x = \frac{59 + \sqrt{1969}}{14}, \frac{59 - \sqrt{1969}}{14}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = \frac{59 + \sqrt{1969}}{14}, \frac{59 - \sqrt{1969}}{14}$

Decimal Form:

$x = 7.38381539 \dots, 1.04475603 \dots$