Find the Intersection of the Functions f(x)=5x^2 , f(x)=x-6

$f (x) = 5 x^{2}$ , $f (x) = x - 6$

Step 1

Substitute $x - 6$ for $f (x)$ .

$x - 6 = 5 x^{2}$

Step 2

Solve for $x$ .

Subtract $5 x^{2}$ from both sides of the equation.

$x - 6 - 5 x^{2} = 0$

Use the quadratic formula to find the solutions.

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

Substitute the values $a = - 5$ , $b = 1$ , and $c = - 6$ into the quadratic formula and solve for $x$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (- 5 \cdot - 6)}}{2 \cdot - 5}$

Simplify.

Simplify the numerator.

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot - 5 \cdot - 6}}{2 \cdot - 5}$

Multiply $- 4 \cdot - 5 \cdot - 6$ .

Multiply $- 4$ by $- 5$ .

$x = \frac{- 1 \pm \sqrt{1 + 20 \cdot - 6}}{2 \cdot - 5}$

Multiply $20$ by $- 6$ .

$x = \frac{- 1 \pm \sqrt{1 - 120}}{2 \cdot - 5}$

Subtract $120$ from $1$ .

$x = \frac{- 1 \pm \sqrt{- 119}}{2 \cdot - 5}$

Rewrite $- 119$ as $- 1 (119)$ .

$x = \frac{- 1 \pm \sqrt{- 1 \cdot 119}}{2 \cdot - 5}$

Rewrite $\sqrt{- 1 (119)}$ as $\sqrt{- 1} \cdot \sqrt{119}$ .

$x = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{119}}{2 \cdot - 5}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 1 \pm i \sqrt{119}}{2 \cdot - 5}$

Multiply $2$ by $- 5$ .

$x = \frac{- 1 \pm i \sqrt{119}}{- 10}$

Simplify $\frac{- 1 \pm i \sqrt{119}}{- 10}$ .

$x = \frac{1 \pm i \sqrt{119}}{10}$

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

Simplify the numerator.

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot - 5 \cdot - 6}}{2 \cdot - 5}$

Multiply $- 4 \cdot - 5 \cdot - 6$ .

Multiply $- 4$ by $- 5$ .

$x = \frac{- 1 \pm \sqrt{1 + 20 \cdot - 6}}{2 \cdot - 5}$

Multiply $20$ by $- 6$ .

$x = \frac{- 1 \pm \sqrt{1 - 120}}{2 \cdot - 5}$

Subtract $120$ from $1$ .

$x = \frac{- 1 \pm \sqrt{- 119}}{2 \cdot - 5}$

Rewrite $- 119$ as $- 1 (119)$ .

$x = \frac{- 1 \pm \sqrt{- 1 \cdot 119}}{2 \cdot - 5}$

Rewrite $\sqrt{- 1 (119)}$ as $\sqrt{- 1} \cdot \sqrt{119}$ .

$x = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{119}}{2 \cdot - 5}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 1 \pm i \sqrt{119}}{2 \cdot - 5}$

Multiply $2$ by $- 5$ .

$x = \frac{- 1 \pm i \sqrt{119}}{- 10}$

Simplify $\frac{- 1 \pm i \sqrt{119}}{- 10}$ .

$x = \frac{1 \pm i \sqrt{119}}{10}$

Change the $\pm$ to $+$ .

$x = \frac{1 + i \sqrt{119}}{10}$

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

Simplify the numerator.

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot - 5 \cdot - 6}}{2 \cdot - 5}$

Multiply $- 4 \cdot - 5 \cdot - 6$ .

Multiply $- 4$ by $- 5$ .

$x = \frac{- 1 \pm \sqrt{1 + 20 \cdot - 6}}{2 \cdot - 5}$

Multiply $20$ by $- 6$ .

$x = \frac{- 1 \pm \sqrt{1 - 120}}{2 \cdot - 5}$

Subtract $120$ from $1$ .

$x = \frac{- 1 \pm \sqrt{- 119}}{2 \cdot - 5}$

Rewrite $- 119$ as $- 1 (119)$ .

$x = \frac{- 1 \pm \sqrt{- 1 \cdot 119}}{2 \cdot - 5}$

Rewrite $\sqrt{- 1 (119)}$ as $\sqrt{- 1} \cdot \sqrt{119}$ .

$x = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{119}}{2 \cdot - 5}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 1 \pm i \sqrt{119}}{2 \cdot - 5}$

Multiply $2$ by $- 5$ .

$x = \frac{- 1 \pm i \sqrt{119}}{- 10}$

Simplify $\frac{- 1 \pm i \sqrt{119}}{- 10}$ .

$x = \frac{1 \pm i \sqrt{119}}{10}$

Change the $\pm$ to $-$ .

$x = \frac{1 - i \sqrt{119}}{10}$

The final answer is the combination of both solutions.

$x = \frac{1 + i \sqrt{119}}{10}, \frac{1 - i \sqrt{119}}{10}$

Find the Intersection of the Functions f(x)=5x^2 , f(x)=x-6

Related Questions

Support LCMGCF.com by disabling your adblocker.