SOLUTION: Consider the quadratic equation ax^2+10x+(a-5) Find the values for a for which there are 1) two solutions 2) no solutions 3) 1 solution

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons  -> Quadratic Equation Lesson -> SOLUTION: Consider the quadratic equation ax^2+10x+(a-5) Find the values for a for which there are 1) two solutions 2) no solutions 3) 1 solution      Log On


   

Question 1136708: Consider the quadratic equation ax^2+10x+(a-5)
Find the values for a for which there are
1) two solutions
2) no solutions
3) 1 solution
Found 2 solutions by MathLover1, Alan3354:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
Consider the quadratic equation ax%5E2%2B10x%2B%28a-5%29
Find the values for a for which there are

1) two solutions
2) no solutions (no real solution)
3) 1 solution (a repeated real number solution)
use discriminant: b%5E2+-+4ac
if discriminant >+0 then 2 solutions
if discriminant = 0+then 1+solution
if discriminant < 0+then no real solution
b%5E2+-+4ac+%3E+0
in your case ax%5E2%2B10x%2B%28a-5%29 , a=a, b=10, c=%28a-5%29
10%5E2+-+4a%28a-5%29+%3E+0
100+-+4a%5E2-20a+%3E+0
100++%3E+4a%5E2-20
4a%5E2%2B20a-100%3C0
a%5E2%2B5a-25%3C0.....use quadratic formula to get
=>%281%2F2%29+%285+-+5+sqrt%285%29%29%3Ca%3C%281%2F2%29+%285+%2B+5+sqrt%285%29%29
=>-3.09%3Ca%3C8.09
check the graph using a=-3
-3x%5E2%2B10x%2B%28-3-5%29
=-3x%5E2%2B10x-8

+graph%28+600%2C+600%2C+-6%2C+5%2C+-10%2C+10%2C+-3x%5E2%2B10x-8%29+
so, there are two solutions


10%5E2+-+4a%28a-5%29+=+0
100+-+4a%5E2-20a+=+0
100++=+4a%5E2-20
4a%5E2%2B20a-100=0
a%5E2%2B5a-25=0.....use quadratic formula to get
=>a=%281%2F2%29+%285+-+5+sqrt%285%29%29 or a=%281%2F2%29+%285+%2B+5+sqrt%285%29%29
=>a=-3.09 or a=8.09
check the graph using a=-3.09
-3.09x%5E2%2B10x%2B%28-3.09-5%29
=-3.09x%5E2%2B10x-8.09

+graph%28+600%2C+600%2C+-6%2C+5%2C+-10%2C+10%2C+-3.09x%5E2%2B10x-8.09%29+


so, there is one solution

10%5E2+-+4a%28a-5%29+%3C+0
100+-+4a%5E2-20a+%3C0
100++%3C+4a%5E2-20
4a%5E2%2B20a-100%3E0
a%5E2%2B5a-25%3E0.....use quadratic formula to get
=>a%3E%281%2F2%29+%285+%2B5+sqrt%285%29%29 or a%3C%281%2F2%29+%285+-+5+sqrt%285%29%29
=>a%3E8.09 or a%3E-3.09
check the graph using a=9
9x%5E2%2B10x%2B%289-5%29
=9x%5E2%2B10x%2B4

+graph%28+600%2C+600%2C+-6%2C+5%2C+-10%2C+10%2C+9x%5E2%2B10x%2B4%29+


so, there are no real solutions







Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
No equal sign --> not an equation.