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)`

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