Finite Math Examples

$- \frac{3}{5} \cdot {(x - 1)}^{2} - 3$

Step 1

Set $- \frac{3}{5} \cdot {(x - 1)}^{2} - 3$ equal to $0$ .

$- \frac{3}{5} \cdot {(x - 1)}^{2} - 3 = 0$

Step 2

Simplify the left side.

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Step 2.1

Simplify each term.

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Step 2.1.1

Combine ${(x - 1)}^{2}$ and $\frac{3}{5}$ .

$- \frac{{(x - 1)}^{2} \cdot 3}{5} - 3 = 0$

Step 2.1.2

Move $3$ to the left of ${(x - 1)}^{2}$ .

$- \frac{3 {(x - 1)}^{2}}{5} - 3 = 0$

Step 3

Multiply through by the least common denominator $5$ , then simplify.

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Step 3.1

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$5 (- \frac{3 {(x - 1)}^{2}}{5} - 3 \cdot \frac{5}{5}) = 0$

Step 3.2

Combine $- 3$ and $\frac{5}{5}$ .

$5 (- \frac{3 {(x - 1)}^{2}}{5} + \frac{- 3 \cdot 5}{5}) = 0$

Step 3.3

Combine the numerators over the common denominator.

$5 (\frac{- 3 {(x - 1)}^{2} - 3 \cdot 5}{5}) = 0$

Step 3.4

Cancel the common factor of $5$ .

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Step 3.4.1

Cancel the common factor.

$5 (\frac{- 3 {(x - 1)}^{2} - 3 \cdot 5}{5}) = 0$

Step 3.4.2

Rewrite the expression.

$- 3 {(x - 1)}^{2} - 3 \cdot 5 = 0$

Step 3.5

Simplify each term.

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Step 3.5.1

Rewrite ${(x - 1)}^{2}$ as $(x - 1) (x - 1)$ .

$- 3 ((x - 1) (x - 1)) - 3 \cdot 5 = 0$

Step 3.5.2

Expand $(x - 1) (x - 1)$ using the FOIL Method.

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Step 3.5.2.1

Apply the distributive property.

$- 3 (x (x - 1) - 1 (x - 1)) - 3 \cdot 5 = 0$

Step 3.5.2.2

Apply the distributive property.

$- 3 (x \cdot x + x \cdot - 1 - 1 (x - 1)) - 3 \cdot 5 = 0$

Step 3.5.2.3

Apply the distributive property.

$- 3 (x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1) - 3 \cdot 5 = 0$

Step 3.5.3

Simplify and combine like terms.

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Step 3.5.3.1

Simplify each term.

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Step 3.5.3.1.1

Multiply $x$ by $x$ .

$- 3 (x^{2} + x \cdot - 1 - 1 x - 1 \cdot - 1) - 3 \cdot 5 = 0$

Step 3.5.3.1.2

Move $- 1$ to the left of $x$ .

$- 3 (x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1) - 3 \cdot 5 = 0$

Step 3.5.3.1.3

Rewrite $- 1 x$ as $- x$ .

$- 3 (x^{2} - x - 1 x - 1 \cdot - 1) - 3 \cdot 5 = 0$

Step 3.5.3.1.4

Rewrite $- 1 x$ as $- x$ .

$- 3 (x^{2} - x - x - 1 \cdot - 1) - 3 \cdot 5 = 0$

Step 3.5.3.1.5

Multiply $- 1$ by $- 1$ .

$- 3 (x^{2} - x - x + 1) - 3 \cdot 5 = 0$

Step 3.5.3.2

Subtract $x$ from $- x$ .

$- 3 (x^{2} - 2 x + 1) - 3 \cdot 5 = 0$

Step 3.5.4

Apply the distributive property.

$- 3 x^{2} - 3 (- 2 x) - 3 \cdot 1 - 3 \cdot 5 = 0$

Step 3.5.5

Simplify.

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Step 3.5.5.1

Multiply $- 2$ by $- 3$ .

$- 3 x^{2} + 6 x - 3 \cdot 1 - 3 \cdot 5 = 0$

Step 3.5.5.2

Multiply $- 3$ by $1$ .

$- 3 x^{2} + 6 x - 3 - 3 \cdot 5 = 0$

Step 3.5.6

Multiply $- 3$ by $5$ .

$- 3 x^{2} + 6 x - 3 - 15 = 0$

Step 3.6

Subtract $15$ from $- 3$ .

$- 3 x^{2} + 6 x - 18 = 0$

Step 4

Use the quadratic formula to find the solutions.

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

Step 5

Substitute the values $a = - 3$ , $b = 6$ , and $c = - 18$ into the quadratic formula and solve for $x$ .

$\frac{- 6 \pm \sqrt{6^{2} - 4 \cdot (- 3 \cdot - 18)}}{2 \cdot - 3}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $6$ to the power of $2$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot - 3 \cdot - 18}}{2 \cdot - 3}$

Step 6.1.2

Multiply $- 4 \cdot - 3 \cdot - 18$ .

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Step 6.1.2.1

Multiply $- 4$ by $- 3$ .

$x = \frac{- 6 \pm \sqrt{36 + 12 \cdot - 18}}{2 \cdot - 3}$

Step 6.1.2.2

Multiply $12$ by $- 18$ .

$x = \frac{- 6 \pm \sqrt{36 - 216}}{2 \cdot - 3}$

Step 6.1.3

Subtract $216$ from $36$ .

$x = \frac{- 6 \pm \sqrt{- 180}}{2 \cdot - 3}$

Step 6.1.4

Rewrite $- 180$ as $- 1 (180)$ .

$x = \frac{- 6 \pm \sqrt{- 1 \cdot 180}}{2 \cdot - 3}$

Step 6.1.5

Rewrite $\sqrt{- 1 (180)}$ as $\sqrt{- 1} \cdot \sqrt{180}$ .

$x = \frac{- 6 \pm \sqrt{- 1} \cdot \sqrt{180}}{2 \cdot - 3}$

Step 6.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 6 \pm i \cdot \sqrt{180}}{2 \cdot - 3}$

Step 6.1.7

Rewrite $180$ as $6^{2} \cdot 5$ .

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Step 6.1.7.1

Factor $36$ out of $180$ .

$x = \frac{- 6 \pm i \cdot \sqrt{36 (5)}}{2 \cdot - 3}$

Step 6.1.7.2

Rewrite $36$ as $6^{2}$ .

$x = \frac{- 6 \pm i \cdot \sqrt{6^{2} \cdot 5}}{2 \cdot - 3}$

Step 6.1.8

Pull terms out from under the radical.

$x = \frac{- 6 \pm i \cdot (6 \sqrt{5})}{2 \cdot - 3}$

Step 6.1.9

Move $6$ to the left of $i$ .

$x = \frac{- 6 \pm 6 i \sqrt{5}}{2 \cdot - 3}$

Step 6.2

Multiply $2$ by $- 3$ .

$x = \frac{- 6 \pm 6 i \sqrt{5}}{- 6}$

Step 6.3

Simplify $\frac{- 6 \pm 6 i \sqrt{5}}{- 6}$ .

$x = 1 \pm i \sqrt{5}$