In the quadratic formula `x=frac(-b ±sqrt(b^2-4ac))(2a)`, the expression under the square root sign `b^2-4ac` is known as the discriminant.
The discriminant indicates three possibilities. If the discriminant is positive and greater than zero, the solution to th equation will have distinct roots (two roots that are different) which would lead to a graph of the function crossing the `x`-axis twice.
If the discriminant is zero, then there will be equal roots and there will only be one answer. The graph would simply touch the `x`-axis.
If the discriminant is less than zero, then the equation will have complex roots and the graph will not touch or cross the `x`-axis. Complex roots are the subject of advanced mathematics.
A quadratic equation is given by `-x^2 - kx - k = 0`.
What is the value of `k`, given that the solution has equal roots?
The discriminant is `b^2 - 4ac`
Given one root, `(k)^2-4(-1)(-k)=0`
`k^2 - 4k=0`
`k = 4`
Answer: `k=4`
Given that `3kx^2+4kx+k+1=0` has equal roots, what is the value of `x`?
There are equal roots, so the discriminant = 0:
`b^2-4ac=0`
Substituting:
`(4k)^2 - 4(3k)(k+1)=0`
`16k^2-12k^2-12k=0`
`4k^2-12k=0`
`k=3`
Rewrite the equation:
`9x^2+12x+4=0`
Substituting into `frac(-b±sqrt(b2-4ac))(2a)`
(Remember the expression under the square root is zero as there are equal roots)
`-frac(8)(12)=-frac(2)(3)`
Check against the original equation:
`3(3)(-frac(2)(3))^2 + 4(3)(-frac(2)(3)) = (3) + 1=0`
`4 - 8 + 4=0`
Answer: `x=-frac(2)(3)`