quadratic equations?

Question

given a quadratic equation and a linear equation, how do you know the number of ponits in which they intersect?

ex. x^2 - 4x + 2 and y=2x - 10 or y=2x + 10 or y= 2x + 7

pls dont just give yes or no answers to the above, tell me how you solve this as well.

i wanna know the algebraic way to do it

Answer 1

A line and a quadratic equation may intersect in 0, 1, or 2 places. You can find out how many points they intersect in by plotting both curves, but you really need to solve the series in order to find out for sure. Graphically, they intersect in 0 points if the line runs completely above or below the vertex of the parabola. They intersect in 1 point if the line is tangent to the parabola. They intersect in 2 points when the line just runs through the parabola.

I'll work through y = x^2 - 4x + 2 and y = 2x as an example. You take the given value of y = 2x and substitute it into the quadratic, giving you 2x = x^2 - 4x + 2 ==> x^2 - 6x + 2 = 0. This can't be factored, so we must apply the quadratic formula. x = (6 +/- sqrt((-6)^2 - 4*1*2))/2 = (6 +/- sqrt(36 - 8))/2 = (6 +/- sqrt(28))/2 = (6 +/- 2*sqrt(7))/2 = 3 +/- sqrt(7). Now you need to find the corresponding y values. Take the two x values, 3 + sqrt(7) and 3 - sqrt(7), and use them to solve y = 2x. Then check to make sure the ordered pairs also satisfy the quadratic. If they do, you have confirmed that there are two points of intersection. When the quadratic formula gives only one solution (because the discriminant is equal to zero), you only have one intersection. Then the quadratic formula gives only imaginary solutions (because the discriminant is negative), you do not have any intersections.

Answer 2

plug it in in your calculator

TI 83 or better

go to On, the blue Y= button at the top, and set the following
Y1=x^2-4x+2 (use the ALPHA key to make letters)
Y2=2x-10
hit the blue GRAPH button at the top

hit the WINDOW button (also blue at the top) and make your settings read...
Xmin= -5
Xmax=10
Xsc1=1
Ymin=-5
Ymax=10
Ysc1=1
Xres=1

and go back to the graph button

you can see that these two objects dont intersect, if they did, then you would go to 2nd, then hit the blue TRACE button (which equals CALC when hit with 2nd) and choose intersect.

you would select the parabola first, hit enter
then select the line and hit enter, then make a guess where the intersection is and hit enter, and it would give you the approrpiate coordinates. you would then trace a sketch of the graph on paper as work shown.

alex