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

$4$ . Solve the following system of equations. (Hint: Use the quadratic formula.) $\newline$ $\begin{array}{l} f(x)=2 x^{2}-3 x-10 \\ g(x)=-3 x^{2}+20 \end{array}$ $\newline$ $(-2.2,5.9)$ 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 $(2.8,-3.0)$

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Math Problems

Algebra 2

Quadratic equation with complex roots

Full solution

4

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

\newline

\begin{array}{l} f(x)=2 x^{2}-3 x-10 \\ g(x)=-3 x^{2}+20 \end{array}

\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 points of intersection. $\newline$ $f(x) = g(x)$ $\newline$ $2x^2 - 3x - 10 = -3x^2 + 20$
Combine Like Terms: Step $2$ : Combine like terms. $\newline$ $2x^2 + 3x^2 - 3x - 10 - 20 = 0$ $\newline$ $5x^2 - 3x - 30 = 0$
Use Quadratic Formula: Step $3$ : Use the quadratic formula to solve for $x$ . The quadratic formula is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ . $a = 5$ , $b = -3$ , $c = -30$ $x = \frac{3 \pm \sqrt{(-3)^2 - 4 \cdot 5 \cdot (-30)}}{2 \cdot 5}$
Calculate Discriminant: Step $4$ : Calculate the discriminant. $\newline$ Discriminant = $(-3)^2 - 4 \cdot 5 \cdot (-30) = 9 + 600 = 609$ $\newline$ $x = \frac{3 \pm \sqrt{609}}{10}$
Simplify Solutions for $x$ : Step $5$ : Simplify the solutions for $x$ . $x = \frac{3 + \sqrt{609}}{10}$ and $x = \frac{3 - \sqrt{609}}{10}$
Substitute Values for y: Step $6$ : Substitute the values of $x$ back into either $f(x)$ or $g(x)$ to find $y$ . Using $f(x)$ : $y_1 = 2\left(\frac{3 + \sqrt{609}}{10}\right)^2 - 3\left(\frac{3 + \sqrt{609}}{10}\right) - 10$ $y_2 = 2\left(\frac{3 - \sqrt{609}}{10}\right)^2 - 3\left(\frac{3 - \sqrt{609}}{10}\right) - 10$
Check Given Points: Step $7$ : Check if the given points $(-2.2,5.9)$ , $(3.2,0.9)$ , $(0,-10)$ , $(1,17)$ , $(2.8,-3.0)$ , $(-1.5,-1)$ , $(-2.2,5.9)$ , and $(2.8,-3.0)$ satisfy either $f(x)$ or $g(x)$ . For $(-2.2,5.9)$ : $(3.2,0.9)$ $1$ $(3.2,0.9)$ $2$