Evaluate Using the Given Value x^2-kx+15=0 , 1/2

$x^{2} - k x + 15 = 0$ , $\frac{1}{2}$

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 $c = 15$ into the quadratic formula and solve for $x$ .

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

Step 3

Simplify.

Simplify the numerator.

Apply the product rule to $- k$ .

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

Raise $- 1$ to the power of $2$ .

$x = \frac{k \pm \sqrt{1 k^{2} - 4 \cdot 1 \cdot 15}}{2 \cdot 1}$

Multiply $k^{2}$ by $1$ .

$x = \frac{k \pm \sqrt{k^{2} - 4 \cdot 1 \cdot 15}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 15$ .

Multiply $- 4$ by $1$ .

$x = \frac{k \pm \sqrt{k^{2} - 4 \cdot 15}}{2 \cdot 1}$

Multiply $- 4$ by $15$ .

$x = \frac{k \pm \sqrt{k^{2} - 60}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{k \pm \sqrt{k^{2} - 60}}{2}$

Step 4

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

Simplify the numerator.

Apply the product rule to $- k$ .

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

Raise $- 1$ to the power of $2$ .

$x = \frac{k \pm \sqrt{1 k^{2} - 4 \cdot 1 \cdot 15}}{2 \cdot 1}$

Multiply $k^{2}$ by $1$ .

$x = \frac{k \pm \sqrt{k^{2} - 4 \cdot 1 \cdot 15}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 15$ .

Multiply $- 4$ by $1$ .

$x = \frac{k \pm \sqrt{k^{2} - 4 \cdot 15}}{2 \cdot 1}$

Multiply $- 4$ by $15$ .

$x = \frac{k \pm \sqrt{k^{2} - 60}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{k \pm \sqrt{k^{2} - 60}}{2}$

Change the $\pm$ to $+$ .

$x = \frac{k + \sqrt{k^{2} - 60}}{2}$

Step 5

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

Simplify the numerator.

Apply the product rule to $- k$ .

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

Raise $- 1$ to the power of $2$ .

$x = \frac{k \pm \sqrt{1 k^{2} - 4 \cdot 1 \cdot 15}}{2 \cdot 1}$

Multiply $k^{2}$ by $1$ .

$x = \frac{k \pm \sqrt{k^{2} - 4 \cdot 1 \cdot 15}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 15$ .

Multiply $- 4$ by $1$ .

$x = \frac{k \pm \sqrt{k^{2} - 4 \cdot 15}}{2 \cdot 1}$

Multiply $- 4$ by $15$ .

$x = \frac{k \pm \sqrt{k^{2} - 60}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{k \pm \sqrt{k^{2} - 60}}{2}$

Change the $\pm$ to $-$ .

$x = \frac{k - \sqrt{k^{2} - 60}}{2}$

Step 6

The final answer is the combination of both solutions.

$x = \frac{k + \sqrt{k^{2} - 60}}{2}$

$x = \frac{k - \sqrt{k^{2} - 60}}{2}$

Step 7

Replace the variable $x$ with $\frac{k + \sqrt{k^{2} - 60}}{2}$ in the expression.

$\frac{1}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$

Evaluate Using the Given Value x^2-kx+15=0 , 1/2

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