$4$ . Solve the following system of equations. (Hint: Use the quadratic formula.) $\newline$ $f(x)=2 x^{2}-3 x-10$ $\newline$ oggle bookmark $=-3 x^{2}+20$ $\newline$ $(-2.2,5.9)$ and $(3.2,0.9)$ $\newline$ $(0,-10)$ and $(1,17)$ $\newline$ $(2.8,-3.0)$ and $(-1.5,-1)$ $\newline$ $(-2.2,5.9)$ and $(2.8,-3.0)$

Full solution

4

. Solve the following system of equations. (Hint: Use the quadratic formula.)

\newline

f(x)=2 x^{2}-3 x-10

\newline

oggle bookmark

=-3 x^{2}+20

\newline

(-2.2,5.9)

and

(3.2,0.9)

\newline

(0,-10)

and

(1,17)

\newline

(2.8,-3.0)

and

(-1.5,-1)

\newline

(-2.2,5.9)

and

(2.8,-3.0)

Set Equations Equal: Step $1$ : Set the equations equal to each other to find the $x$ -values where $f(x) = g(x)$ . $\newline$ Calculation: $2x^2 - 3x - 10 = -3x^2 + 20$
Combine Like Terms: Step $2$ : Combine like terms to form a single quadratic equation. $\newline$ Calculation: $2x^2 + 3x^2 - 3x - 10 - 20 = 0$ $\newline$ $5x^2 - 3x - 30 = 0$
Use Quadratic Formula: Step $3$ : Use the quadratic formula, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ , to find the roots. $a = 5$ , $b = -3$ , $c = -30$ Calculation: $x = \frac{3 \pm \sqrt{(-3)^2 - 4\cdot5\cdot(-30)}}{2\cdot5}$
Calculate Discriminant: Step $4$ : Calculate the discriminant $b^2 - 4ac$ . $\newline$ Calculation: $(-3)^2 - 4\cdot5\cdot(-30) = 9 + 600 = 609$
Solve for x: Step $5$ : Solve for x using the values from the quadratic formula. $\newline$ Calculation: $x = \frac{3 \pm \sqrt{609}}{10}$
Simplify Roots: Step $6$ : Simplify the roots. $\newline$ Calculation: $x = \frac{3 + \sqrt{609}}{10}$ and $x = \frac{3 - \sqrt{609}}{10}$
Substitute for y: Step $7$ : Substitute the $x$ -values back into either $f(x)$ or $g(x)$ to find the corresponding $y$ -values. $\newline$ Calculation: For $x = \frac{3 + \sqrt{609}}{10}$ , $y = 2\left(\frac{3 + \sqrt{609}}{10}\right)^2 - 3\left(\frac{3 + \sqrt{609}}{10}\right) - 10$ $\newline$ For $x = \frac{3 - \sqrt{609}}{10}$ , $y = 2\left(\frac{3 - \sqrt{609}}{10}\right)^2 - 3\left(\frac{3 - \sqrt{609}}{10}\right) - 10$