Finite Math Examples

$- x^{2} + 15 x - 13 = 0$

Step 1

Factor $- 1$ out of $- x^{2} + 15 x - 13$ .

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Step 1.1

Factor $- 1$ out of $- x^{2}$ .

$- (x^{2}) + 15 x - 13 = 0$

Step 1.2

Factor $- 1$ out of $15 x$ .

$- (x^{2}) - (- 15 x) - 13 = 0$

Step 1.3

Rewrite $- 13$ as $- 1 (13)$ .

$- (x^{2}) - (- 15 x) - 1 \cdot 13 = 0$

Step 1.4

Factor $- 1$ out of $- (x^{2}) - (- 15 x)$ .

$- (x^{2} - 15 x) - 1 \cdot 13 = 0$

Step 1.5

Factor $- 1$ out of $- (x^{2} - 15 x) - 1 (13)$ .

$- (x^{2} - 15 x + 13) = 0$

Step 2

Use the quadratic formula to find the roots for $x^{2} - 15 x + 13 = 0$

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Step 2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a} = 0$

Step 2.2

Substitute the values $a = 1$ , $b = - 15$ , and $c = 13$ into the quadratic formula and solve for $x$ .

$\frac{15 \pm \sqrt{{(- 15)}^{2} - 4 \cdot (1 \cdot 13)}}{2 \cdot 1} = 0$

Step 2.3

Simplify.

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Step 2.3.1

Simplify the numerator.

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Step 2.3.1.1

Raise $- 15$ to the power of $2$ .

$x = \frac{15 \pm \sqrt{225 - 4 \cdot 1 \cdot 13}}{2 \cdot 1}$

Step 2.3.1.2

Multiply $- 4 \cdot 1 \cdot 13$ .

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Step 2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{15 \pm \sqrt{225 - 4 \cdot 13}}{2 \cdot 1}$

Step 2.3.1.2.2

Multiply $- 4$ by $13$ .

$x = \frac{15 \pm \sqrt{225 - 52}}{2 \cdot 1}$

Step 2.3.1.3

Subtract $52$ from $225$ .

$x = \frac{15 \pm \sqrt{173}}{2 \cdot 1}$

Step 2.3.2

Multiply $2$ by $1$ .

$x = \frac{15 \pm \sqrt{173}}{2}$

Step 2.4

The final answer is the combination of both solutions.

$x = \frac{15 + \sqrt{173}}{2}, \frac{15 - \sqrt{173}}{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = \frac{15 + \sqrt{173}}{2}, \frac{15 - \sqrt{173}}{2}$

Decimal Form:

$x = 14.07647321 \dots, 0.92352678 \dots$