# Maths - Discriminant

> Source: https://ollybritton.com/notes/a-level/maths/topics/discriminant/ · Updated: 2020-10-24 · Tags: maths

##### Where does the value of $b^2 - 4ac$ come from for the discriminant??
The value under the square root sign in the quadratic formula.

##### Why is the discriminant useful??
It tells you how many real roots a quadratic has.

##### What is the formula for the discriminant??
$$
b^2 - 4ac
$$

##### Why must a positive discriminant mean two real roots??
The square root of a positive number has two possible solutions.

##### Why must a zero discriminant mean one repeated root??
The square root of zero is zero; there is only one solution.

##### Why must a negative discriminant mean no real roots??
Because the square root of a negative number is imaginary.

##### How many real solutions for $b^2 - 4ac > 0$??
$$
2
$$

##### How many real solutions for $b^2 - 4ac = 0$??
$$
1
$$

##### How many real solutions for $b^2 - 4ac < 0$??
$$
0
$$

##### How would you go about finding the values for $k$ for which $f(x) = x^2 + kx + 9$ has equal roots??
Use the fact that $b^2 - 4ac$ must equal $0$ and substitute for $k$.

##### Given that $y - x = k$ and $x^2 + y^2 = 4$ has one pair of solutions, what two steps could you use to show that $k = \pm2\sqrt{2}$??
* Substitute $x = k + y$ into $x^2 + y^2 = 4$ to find a quadratic in terms of $k$ and $y$.
* Use the fact the discriminant has to equal $0$ for there to be one pair of solutions.

##### If you have something like $4kx^2 + 2x + k + 1$, even if the discriminant is positive why does a value of $k = 0$ still mean there is only $1$ solution??
Because the $x^2$ term disappears and you get a linear equation with only one root.

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