Write as a Function of k x^2-kx+100=0

$x^{2} - k x + 100 = 0$

Step 1

Use the quadratic formula to find the solutions.

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

Step 2

Substitute the values $a = 1$ , $b = - k$ , and $c = 100$ into the quadratic formula and solve for $x$ .

$\frac{k \pm \sqrt{{(- k)}^{2} - 4 \cdot (1 \cdot 100)}}{2 \cdot 1}$

Step 3

Simplify.

Simplify the numerator.

Rewrite $4 \cdot 1 \cdot 100$ as ${(2 \cdot 1 \cdot 10)}^{2}$ .

$x = \frac{k \pm \sqrt{{(- k)}^{2} - {(2 \cdot 1 \cdot 10)}^{2}}}{2 \cdot 1}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = - k$ and $b = 2 \cdot 1 \cdot 10$ .

$x = \frac{k \pm \sqrt{(- k + 2 \cdot 1 \cdot 10) (- k - (2 \cdot 1 \cdot 10))}}{2 \cdot 1}$

Simplify.

Multiply $2 \cdot 1 \cdot 10$ .

Multiply $2$ by $1$ .

$x = \frac{k \pm \sqrt{(- k + 2 \cdot 10) (- k - (2 \cdot 1 \cdot 10))}}{2 \cdot 1}$

Multiply $2$ by $10$ .

$x = \frac{k \pm \sqrt{(- k + 20) (- k - (2 \cdot 1 \cdot 10))}}{2 \cdot 1}$

Multiply $- (2 \cdot 1 \cdot 10)$ .

Multiply $2$ by $1$ .

$x = \frac{k \pm \sqrt{(- k + 20) (- k - (2 \cdot 10))}}{2 \cdot 1}$

Multiply $2$ by $10$ .

$x = \frac{k \pm \sqrt{(- k + 20) (- k - 1 \cdot 20)}}{2 \cdot 1}$

Multiply $- 1$ by $20$ .

$x = \frac{k \pm \sqrt{(- k + 20) (- k - 20)}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{k \pm \sqrt{(- k + 20) (- k - 20)}}{2}$

Step 4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Simplify the numerator.

Rewrite $4 \cdot 1 \cdot 100$ as ${(2 \cdot 1 \cdot 10)}^{2}$ .

$x = \frac{k \pm \sqrt{{(- k)}^{2} - {(2 \cdot 1 \cdot 10)}^{2}}}{2 \cdot 1}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = - k$ and $b = 2 \cdot 1 \cdot 10$ .

$x = \frac{k \pm \sqrt{(- k + 2 \cdot 1 \cdot 10) (- k - (2 \cdot 1 \cdot 10))}}{2 \cdot 1}$

Simplify.

Multiply $2 \cdot 1 \cdot 10$ .

Multiply $2$ by $1$ .

$x = \frac{k \pm \sqrt{(- k + 2 \cdot 10) (- k - (2 \cdot 1 \cdot 10))}}{2 \cdot 1}$

Multiply $2$ by $10$ .

$x = \frac{k \pm \sqrt{(- k + 20) (- k - (2 \cdot 1 \cdot 10))}}{2 \cdot 1}$

Multiply $- (2 \cdot 1 \cdot 10)$ .

Multiply $2$ by $1$ .

$x = \frac{k \pm \sqrt{(- k + 20) (- k - (2 \cdot 10))}}{2 \cdot 1}$

Multiply $2$ by $10$ .

$x = \frac{k \pm \sqrt{(- k + 20) (- k - 1 \cdot 20)}}{2 \cdot 1}$

Multiply $- 1$ by $20$ .

$x = \frac{k \pm \sqrt{(- k + 20) (- k - 20)}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{k \pm \sqrt{(- k + 20) (- k - 20)}}{2}$

Change the $\pm$ to $+$ .

$x = \frac{k + \sqrt{(- k + 20) (- k - 20)}}{2}$

Step 5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Simplify the numerator.

Rewrite $4 \cdot 1 \cdot 100$ as ${(2 \cdot 1 \cdot 10)}^{2}$ .

$x = \frac{k \pm \sqrt{{(- k)}^{2} - {(2 \cdot 1 \cdot 10)}^{2}}}{2 \cdot 1}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = - k$ and $b = 2 \cdot 1 \cdot 10$ .

$x = \frac{k \pm \sqrt{(- k + 2 \cdot 1 \cdot 10) (- k - (2 \cdot 1 \cdot 10))}}{2 \cdot 1}$

Simplify.

Multiply $2 \cdot 1 \cdot 10$ .

Multiply $2$ by $1$ .

$x = \frac{k \pm \sqrt{(- k + 2 \cdot 10) (- k - (2 \cdot 1 \cdot 10))}}{2 \cdot 1}$

Multiply $2$ by $10$ .

$x = \frac{k \pm \sqrt{(- k + 20) (- k - (2 \cdot 1 \cdot 10))}}{2 \cdot 1}$

Multiply $- (2 \cdot 1 \cdot 10)$ .

Multiply $2$ by $1$ .

$x = \frac{k \pm \sqrt{(- k + 20) (- k - (2 \cdot 10))}}{2 \cdot 1}$

Multiply $2$ by $10$ .

$x = \frac{k \pm \sqrt{(- k + 20) (- k - 1 \cdot 20)}}{2 \cdot 1}$

Multiply $- 1$ by $20$ .

$x = \frac{k \pm \sqrt{(- k + 20) (- k - 20)}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{k \pm \sqrt{(- k + 20) (- k - 20)}}{2}$

Change the $\pm$ to $-$ .

$x = \frac{k - \sqrt{(- k + 20) (- k - 20)}}{2}$

Step 6

The final answer is the combination of both solutions.

$x = \frac{k + \sqrt{(- k + 20) (- k - 20)}}{2}$

$x = \frac{k - \sqrt{(- k + 20) (- k - 20)}}{2}$

Write as a Function of k x^2-kx+100=0

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