Finite Math Examples

$k x^{2} = k + x$

Step 1

Subtract $x$ from both sides of the equation.

$k x^{2} - x = k$

Step 2

Subtract $k$ from both sides of the equation.

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

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

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

Step 5

Simplify the numerator.

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Step 5.1

Factor $- 1$ out of ${(- 1)}^{2} - 4 \cdot k \cdot (- k)$ .

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Step 5.1.1

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

$x = \frac{1 \pm \sqrt{1 - 4 \cdot k \cdot (- k)}}{2 k}$

Step 5.1.2

Factor $- 1$ out of $- 4 \cdot k \cdot (- k)$ .

$x = \frac{1 \pm \sqrt{1 - (- 4 \cdot k \cdot k)}}{2 k}$

Step 5.1.3

Factor $- 1$ out of $- - 1 - (- 4 \cdot k \cdot (k))$ .

$x = \frac{1 \pm \sqrt{- (- 1 - 4 \cdot k \cdot k)}}{2 k}$

Step 5.2

Multiply $k$ by $k$ by adding the exponents.

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Step 5.2.1

Move $k$ .

$x = \frac{1 \pm \sqrt{- (- 1 - 4 \cdot (k \cdot k))}}{2 k}$

Step 5.2.2

Multiply $k$ by $k$ .

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

$x = \frac{1 \pm \sqrt{- (- 1 - 4 k^{2})}}{2 k}$

Step 6

The final answer is the combination of both solutions.

$x = \frac{1 + \sqrt{- (- 1 - 4 k^{2})}}{2 k}$

$x = \frac{1 - \sqrt{- (- 1 - 4 k^{2})}}{2 k}$