Finite Math Examples

$0.75 x^{2} - x + 400 = 8.75$

Step 1

Subtract $8.75$ from both sides of the equation.

$0.75 x^{2} - x + 400 - 8.75 = 0$

Step 2

Subtract $8.75$ from $400$ .

$0.75 x^{2} - x + 391.25 = 0$

Step 3

Factor $0.25$ out of $0.75 x^{2} - x + 391.25$ .

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Step 3.1

Factor $0.25$ out of $0.75 x^{2}$ .

$0.25 (3 x^{2}) - x + 391.25 = 0$

Step 3.2

Factor $0.25$ out of $- x$ .

$0.25 (3 x^{2}) + 0.25 (- 4 x) + 391.25 = 0$

Step 3.3

Factor $0.25$ out of $391.25$ .

$0.25 (3 x^{2}) + 0.25 (- 4 x) + 0.25 (1565) = 0$

Step 3.4

Factor $0.25$ out of $0.25 (3 x^{2}) + 0.25 (- 4 x)$ .

$0.25 (3 x^{2} - 4 x) + 0.25 (1565) = 0$

Step 3.5

Factor $0.25$ out of $0.25 (3 x^{2} - 4 x) + 0.25 (1565)$ .

$0.25 (3 x^{2} - 4 x + 1565) = 0$

Step 4

Divide each term in $0.25 (3 x^{2} - 4 x + 1565) = 0$ by $0.25$ and simplify.

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Step 4.1

Divide each term in $0.25 (3 x^{2} - 4 x + 1565) = 0$ by $0.25$ .

$\frac{0.25 (3 x^{2} - 4 x + 1565)}{0.25} = \frac{0}{0.25}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $0.25$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{0.25 (3 x^{2} - 4 x + 1565)}{0.25} = \frac{0}{0.25}$

Step 4.2.1.2

Divide $3 x^{2} - 4 x + 1565$ by $1$ .

$3 x^{2} - 4 x + 1565 = \frac{0}{0.25}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $0$ by $0.25$ .

$3 x^{2} - 4 x + 1565 = 0$

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 $a = 3$ , $b = - 4$ , and $c = 1565$ into the quadratic formula and solve for $x$ .

$\frac{4 \pm \sqrt{{(- 4)}^{2} - 4 \cdot (3 \cdot 1565)}}{2 \cdot 3}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $- 4$ to the power of $2$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 3 \cdot 1565}}{2 \cdot 3}$

Step 7.1.2

Multiply $- 4 \cdot 3 \cdot 1565$ .

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Step 7.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{4 \pm \sqrt{16 - 12 \cdot 1565}}{2 \cdot 3}$

Step 7.1.2.2

Multiply $- 12$ by $1565$ .

$x = \frac{4 \pm \sqrt{16 - 18780}}{2 \cdot 3}$

Step 7.1.3

Subtract $18780$ from $16$ .

$x = \frac{4 \pm \sqrt{- 18764}}{2 \cdot 3}$

Step 7.1.4

Rewrite $- 18764$ as $- 1 (18764)$ .

$x = \frac{4 \pm \sqrt{- 1 \cdot 18764}}{2 \cdot 3}$

Step 7.1.5

Rewrite $\sqrt{- 1 (18764)}$ as $\sqrt{- 1} \cdot \sqrt{18764}$ .

$x = \frac{4 \pm \sqrt{- 1} \cdot \sqrt{18764}}{2 \cdot 3}$

Step 7.1.6

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

$x = \frac{4 \pm i \cdot \sqrt{18764}}{2 \cdot 3}$

Step 7.1.7

Rewrite $18764$ as $2^{2} \cdot 4691$ .

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Step 7.1.7.1

Factor $4$ out of $18764$ .

$x = \frac{4 \pm i \cdot \sqrt{4 (4691)}}{2 \cdot 3}$

Step 7.1.7.2

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

$x = \frac{4 \pm i \cdot \sqrt{2^{2} \cdot 4691}}{2 \cdot 3}$

Step 7.1.8

Pull terms out from under the radical.

$x = \frac{4 \pm i \cdot (2 \sqrt{4691})}{2 \cdot 3}$

Step 7.1.9

Move $2$ to the left of $i$ .

$x = \frac{4 \pm 2 i \sqrt{4691}}{2 \cdot 3}$

Step 7.2

Multiply $2$ by $3$ .

$x = \frac{4 \pm 2 i \sqrt{4691}}{6}$

Step 7.3

Simplify $\frac{4 \pm 2 i \sqrt{4691}}{6}$ .

$x = \frac{2 \pm i \sqrt{4691}}{3}$

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 $- 4$ to the power of $2$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 3 \cdot 1565}}{2 \cdot 3}$

Step 8.1.2

Multiply $- 4 \cdot 3 \cdot 1565$ .

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Step 8.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{4 \pm \sqrt{16 - 12 \cdot 1565}}{2 \cdot 3}$

Step 8.1.2.2

Multiply $- 12$ by $1565$ .

$x = \frac{4 \pm \sqrt{16 - 18780}}{2 \cdot 3}$

Step 8.1.3

Subtract $18780$ from $16$ .

$x = \frac{4 \pm \sqrt{- 18764}}{2 \cdot 3}$

Step 8.1.4

Rewrite $- 18764$ as $- 1 (18764)$ .

$x = \frac{4 \pm \sqrt{- 1 \cdot 18764}}{2 \cdot 3}$

Step 8.1.5

Rewrite $\sqrt{- 1 (18764)}$ as $\sqrt{- 1} \cdot \sqrt{18764}$ .

$x = \frac{4 \pm \sqrt{- 1} \cdot \sqrt{18764}}{2 \cdot 3}$

Step 8.1.6

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

$x = \frac{4 \pm i \cdot \sqrt{18764}}{2 \cdot 3}$

Step 8.1.7

Rewrite $18764$ as $2^{2} \cdot 4691$ .

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Step 8.1.7.1

Factor $4$ out of $18764$ .

$x = \frac{4 \pm i \cdot \sqrt{4 (4691)}}{2 \cdot 3}$

Step 8.1.7.2

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

$x = \frac{4 \pm i \cdot \sqrt{2^{2} \cdot 4691}}{2 \cdot 3}$

Step 8.1.8

Pull terms out from under the radical.

$x = \frac{4 \pm i \cdot (2 \sqrt{4691})}{2 \cdot 3}$

Step 8.1.9

Move $2$ to the left of $i$ .

$x = \frac{4 \pm 2 i \sqrt{4691}}{2 \cdot 3}$

Step 8.2

Multiply $2$ by $3$ .

$x = \frac{4 \pm 2 i \sqrt{4691}}{6}$

Step 8.3

Simplify $\frac{4 \pm 2 i \sqrt{4691}}{6}$ .

$x = \frac{2 \pm i \sqrt{4691}}{3}$

Step 8.4

Change the $\pm$ to $+$ .

$x = \frac{2 + i \sqrt{4691}}{3}$

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 $- 4$ to the power of $2$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 3 \cdot 1565}}{2 \cdot 3}$

Step 9.1.2

Multiply $- 4 \cdot 3 \cdot 1565$ .

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Step 9.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{4 \pm \sqrt{16 - 12 \cdot 1565}}{2 \cdot 3}$

Step 9.1.2.2

Multiply $- 12$ by $1565$ .

$x = \frac{4 \pm \sqrt{16 - 18780}}{2 \cdot 3}$

Step 9.1.3

Subtract $18780$ from $16$ .

$x = \frac{4 \pm \sqrt{- 18764}}{2 \cdot 3}$

Step 9.1.4

Rewrite $- 18764$ as $- 1 (18764)$ .

$x = \frac{4 \pm \sqrt{- 1 \cdot 18764}}{2 \cdot 3}$

Step 9.1.5

Rewrite $\sqrt{- 1 (18764)}$ as $\sqrt{- 1} \cdot \sqrt{18764}$ .

$x = \frac{4 \pm \sqrt{- 1} \cdot \sqrt{18764}}{2 \cdot 3}$

Step 9.1.6

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

$x = \frac{4 \pm i \cdot \sqrt{18764}}{2 \cdot 3}$

Step 9.1.7

Rewrite $18764$ as $2^{2} \cdot 4691$ .

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Step 9.1.7.1

Factor $4$ out of $18764$ .

$x = \frac{4 \pm i \cdot \sqrt{4 (4691)}}{2 \cdot 3}$

Step 9.1.7.2

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

$x = \frac{4 \pm i \cdot \sqrt{2^{2} \cdot 4691}}{2 \cdot 3}$

Step 9.1.8

Pull terms out from under the radical.

$x = \frac{4 \pm i \cdot (2 \sqrt{4691})}{2 \cdot 3}$

Step 9.1.9

Move $2$ to the left of $i$ .

$x = \frac{4 \pm 2 i \sqrt{4691}}{2 \cdot 3}$

Step 9.2

Multiply $2$ by $3$ .

$x = \frac{4 \pm 2 i \sqrt{4691}}{6}$

Step 9.3

Simplify $\frac{4 \pm 2 i \sqrt{4691}}{6}$ .

$x = \frac{2 \pm i \sqrt{4691}}{3}$

Step 9.4

Change the $\pm$ to $-$ .

$x = \frac{2 - i \sqrt{4691}}{3}$

Step 10

The final answer is the combination of both solutions.

$x = \frac{2 + i \sqrt{4691}}{3}, \frac{2 - i \sqrt{4691}}{3}$