Systems of Quadratic Equations Question

Question

looking for help on this question. Solve the following systems of equations algebraically using the quadratic formula.

$$\begin{align} y& =-x^2+2x+9\\ y& =-5x^2+10x+12\end{align}$$

Any help would be appreciated!

Answer 1

Hint: put both equations equal to one another. You'll have a quadratic equation.

$$-x^2+2x+9 = -5x^2+10x+12$$

Now simplify (combine "like terms": get all terms on one side (say left hand side) equal to $0$ (on the right-hand side).

Factor and/or use the quadratic formula to find any solution(s), if they exist, and they do exist: there are two solutions for $x$, each of which is a "zero" of the equation.

Answer 2

set $z=-x^2+2x$, we get:

$y=z+9\\y=5z+12$

So, $z=-\frac{3}{4}, y=\frac{33}{4}$

Then you can solve $-\frac{3}{4}=-x^2+2x$.

Answer 3

Note: Because the system you are given has $y$ solved for in both cases, graphing could be an option.

However, this does not excuse one from knowing how to solve the system algebraically.

Solving Graphically: \begin{align} y& =-x^2+2x+9\\ y& =-5x^2+10x+12\end{align} https://www.desmos.com/calculator/lkub3kc0hx

Looking for intersecting points we find $x_1=-.32$ & $x_2=2.32$