Finding the set of values of k

By the end of this lesson

By the end you can find the values of an unknown constant $k$ that give a quadratic two distinct, equal or no real roots.

That usually means solving a quadratic inequality in $k$ .

What is going on here?

Sometimes a quadratic contains a letter like $k$ . It stands for an unknown constant: a fixed number we haven't been told.

So $x^2 + kx + 9 = 0$ is really a whole family of equations, one for each value of $k$ :

Some members of the family have two roots, some one, some none.

The question is always: which values of $k$ give the kind of roots we want?

The discriminant is $b^2 - 4ac$ , the part under the square root in the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

It decides how many roots there are:

Discriminant	What the $\pm$ does	Roots
$b^2 - 4ac > 0$	$\pm$ a positive number gives two different answers	two distinct real roots
$b^2 - 4ac = 0$	$\pm 0$ gives the same answer twice	one repeated root (also called equal roots)
$b^2 - 4ac < 0$	needs the square root of a negative, which isn't real	no real roots

So here's the plan. Instead of solving for $x$ , write the discriminant in terms of $k$ , then solve for $k$ .

Get the equation into the form $ax^2 + bx + c = 0$ .
Write down $a$ , $b$ and $c$ in terms of $k$ . Use brackets.
Turn the wording into a condition on $b^2 - 4ac$ : $> 0$ , $= 0$ , $\geqslant 0$ or $< 0$ .
Simplify. You'll usually get a quadratic in $k$ .
Solve it. An equation if the condition is $= 0$ ; an inequality (with a sketch) otherwise.

These are skills you already have, just stacked on top of each other. Like a lasagne.

Worked example: Example 1: two distinct real roots

Find the set of values of $k$ for which $x^2 + kx + k + 3 = 0$ has two distinct real roots.

Have a go first?

Quick check

Write an expressionQuestion 1 · Recall

CalculateQuestion 2 · Apply

Answer the 2 questions above to continue.