Question

How do you know if a quadratic equation will have one, two, or no solutions?
How do you find a quadratic equation if you are only given the solution?
Is it possible to have different quadratic equations with the same solution?

Answer 1

1) put the equation as
\[ax^2+bx+c=0\]then
\[b^2-4ac\]say it all if
\[b^2-4ac<0\] i.e. is negative, then there is no real solution
if
\[b^2-4ac=0\]then there is one solution and if
\[b^2-4ac>0\] i.e. it is positive, then there are two solutions

Answer 2

2) suppose you are given the solutions as
\[r_1,r_2\] then you can find an equation that has these solutions by writing
\[(x-r_1)(x-r_2)=0\]
for example, if the two solutions are -3 and 5 you would write
\[(x+3)(x-5)=0\] or
\[x^2+2x-15=0\]

Answer 3

typo there, should have been
\[x^2-2x-15=0\] sorry

Answer 4

yes, you can have two different quadratic equations that have the same solution, but one will be a multiple of the other. for example
\[x^2-2x-15=0\] has solutions -3 and 5, but so does
\[-2x^2+4x+30=0\] (i multiplied by -1)

two equations with the same solution are called "equivalent"

Answer 5

Thank you so much!

Answer 6

yw