Finite Math Examples

$4 k^{2} - 92 k + 100 = 0$

Step 1

Factor $4$ out of $4 k^{2} - 92 k + 100$ .

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Step 1.1

Factor $4$ out of $4 k^{2}$ .

$4 (k^{2}) - 92 k + 100 = 0$

Step 1.2

Factor $4$ out of $- 92 k$ .

$4 (k^{2}) + 4 (- 23 k) + 100 = 0$

Step 1.3

Factor $4$ out of $100$ .

$4 k^{2} + 4 (- 23 k) + 4 \cdot 25 = 0$

Step 1.4

Factor $4$ out of $4 k^{2} + 4 (- 23 k)$ .

$4 (k^{2} - 23 k) + 4 \cdot 25 = 0$

Step 1.5

Factor $4$ out of $4 (k^{2} - 23 k) + 4 \cdot 25$ .

$4 (k^{2} - 23 k + 25) = 0$

Step 2

Divide each term in $4 (k^{2} - 23 k + 25) = 0$ by $4$ and simplify.

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Step 2.1

Divide each term in $4 (k^{2} - 23 k + 25) = 0$ by $4$ .

$\frac{4 (k^{2} - 23 k + 25)}{4} = \frac{0}{4}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $4$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{4 (k^{2} - 23 k + 25)}{4} = \frac{0}{4}$

Step 2.2.1.2

Divide $k^{2} - 23 k + 25$ by $1$ .

$k^{2} - 23 k + 25 = \frac{0}{4}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $0$ by $4$ .

$k^{2} - 23 k + 25 = 0$

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = 1$ , $b = - 23$ , and $c = 25$ into the quadratic formula and solve for $k$ .

$\frac{23 \pm \sqrt{{(- 23)}^{2} - 4 \cdot (1 \cdot 25)}}{2 \cdot 1}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $- 23$ to the power of $2$ .

$k = \frac{23 \pm \sqrt{529 - 4 \cdot 1 \cdot 25}}{2 \cdot 1}$

Step 5.1.2

Multiply $- 4 \cdot 1 \cdot 25$ .

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Step 5.1.2.1

Multiply $- 4$ by $1$ .

$k = \frac{23 \pm \sqrt{529 - 4 \cdot 25}}{2 \cdot 1}$

Step 5.1.2.2

Multiply $- 4$ by $25$ .

$k = \frac{23 \pm \sqrt{529 - 100}}{2 \cdot 1}$

Step 5.1.3

Subtract $100$ from $529$ .

$k = \frac{23 \pm \sqrt{429}}{2 \cdot 1}$

Step 5.2

Multiply $2$ by $1$ .

$k = \frac{23 \pm \sqrt{429}}{2}$

Step 6

The final answer is the combination of both solutions.

$k = \frac{23 + \sqrt{429}}{2}, \frac{23 - \sqrt{429}}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$k = \frac{23 + \sqrt{429}}{2}, \frac{23 - \sqrt{429}}{2}$

Decimal Form:

$k = 21.85615758 \dots, 1.14384241 \dots$