SOLUTION: Find the values for a for which the graph y=a^2x^2-2ax-a+1 1) crosses the x-axis 2) does not cross the x-axis

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons  -> Quadratic Equation Lesson -> SOLUTION: Find the values for a for which the graph y=a^2x^2-2ax-a+1 1) crosses the x-axis 2) does not cross the x-axis      Log On


   

Question 1136706: Find the values for a for which the graph y=a^2x^2-2ax-a+1
1) crosses the x-axis
2) does not cross the x-axis
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

y=a%5E2x%5E2-2ax-a%2B1
1) crosses the x-axis
2) does not cross the x-axis

use discriminant D=b^2-4ac
if D%3E0 we have two+distinct+real zeros
if D%3C0 we have two complex zeros

D=%282a%29%5E2-4a%5E2%2A%28-a%2B1%29
D=4a%5E2%2B4a%5E3%2B4

find
4a%5E3%2B4a%5E2%2B4%3E0..simplify
a%5E3%2Ba%5E2%2B1%3E0-> put it in calculator and you will get
a%3E-1.46557
now try with a=1 which is greater then -1.46557
y=1%5E2x%5E2-2%2A1%2Ax-1%2B1
y=x%5E2-2x
find x-intercepts:
x%5E2-2x=0
%28x-2%29x=0
=>x=0 or x=2
check the graph to see x-intercepts
+graph%28+600%2C+600%2C+-10%2C10%2C+-10%2C+10%2C+x%5E2-2x%29+


and

if D%3C0 we have two complex zeros; means there are no x-intercepts
a%3C-1.46557
choose any value of a greater than -1.46557: try with a=-1
y=%28-1%29%5E2x%5E2-2%2A%28-1%29%2Ax-%28-1%29%2B1
y=x%5E2%2B2x%2B1%2B1
y=x%5E2%2B2x%2B2

+graph%28+600%2C+600%2C+-10%2C10%2C+-10%2C+10%2Cx%5E2%2B2x%2B2%29+