SOLUTION: Use the quadratic formula to find any x-intercepts of the parabola. (If an answer does not exist, enter DNE.) y = 4x^2 − 20x + 5

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons -> SOLUTION: Use the quadratic formula to find any x-intercepts of the parabola. (If an answer does not exist, enter DNE.) y = 4x^2 − 20x + 5      Log On


   

Question 710442: Use the quadratic formula to find any x-intercepts of the parabola. (If an answer does not exist, enter DNE.)
y = 4x^2 − 20x + 5
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
For an equation of the form:
ax%5E2%2Bbx%2Bc=0
the Quadratic Formula gives us a way to find solutions for x.

Your equation:
y+=+4x%5E2-20x%2B5
is not quite in this form. But the problem says we are interested in x-intercepts. X-intercepts are where a graph intersects the x-axis. All points on the x-axis have a y coordinate of 0. So to find x-intercepts in general you make the y be zero and then solve for x. So our equation for the x-intercepts will be:
0+=+4x%5E2-20x%2B5
This is closer to the form for the Quadratic Formula. It may be close enough that you can already see what numbers "a", "b" and "c" are. But to be perfectly clear we are going to get the equation to match the form exactly. The form has only additions in it. Our equation has a subtraction. So we must convert the subtraction into an equivalent addition (adding the opposite):
0+=+4x%5E2%2B%28-20x%29%2B5
We now match the form exactly. And we can see that
a = 4
b = -20
c = 5

With the equation in the proper form and with these values for a, b and c we can now use the Quadratic Formula:
x+=+%28-b+%2B-+sqrt%28b%5E2-4ac%29%29%2F2a
Substituting in our values for a, b and c we get:
x+=+%28-%28-20%29+%2B-+sqrt%28%28-20%29%5E2-4%284%29%285%29%29%29%2F2%284%29
Simplifying...
x+=+%28-%28-20%29+%2B-+sqrt%28400-4%284%29%285%29%29%29%2F2%284%29
x+=+%28-%28-20%29+%2B-+sqrt%28400-80%29%29%2F2%284%29
x+=+%28-%28-20%29+%2B-+sqrt%28320%29%29%2F2%284%29
x+=+%2820+%2B-+sqrt%28320%29%29%2F8
x+=+%2820+%2B-+sqrt%2864%2A5%29%29%2F8
x+=+%2820+%2B-+sqrt%2864%29%2Asqrt%285%29%29%2F8
x+=+%2820+%2B-+8%2Asqrt%285%29%29%2F8
x+=+%284%285+%2B-+2%2Asqrt%285%29%29%29%2F%284%2A2%29
x+=+%28cross%284%29%285+%2B-+2%2Asqrt%285%29%29%29%2F%28cross%284%29%2A2%29
leaving:
x+=+%285+%2B-+2%2Asqrt%285%29%29%2F2
which is short for:
x+=+%285+%2B+2%2Asqrt%285%29%29%2F2 or x+=+%285+-+2%2Asqrt%285%29%29%2F2
These are the solutions to our equation.