Using the discriminant

Apply the discriminant to find unknown coefficients, decide whether roots are real, equal or rational, and find when a line is tangent to a parabola.

What the discriminant tells you

Discriminant

$$\Delta = b^2 - 4ac$$

Condition	Meaning
$\Delta > 0$	two distinct real roots
$\Delta = 0$	equal (double) roots
$\Delta \ge 0$	real roots (of either kind)
$\Delta < 0$	no real roots
$\Delta$ a perfect square	rational roots (if $a$, $b$, $c$ are rational)

Finding an unknown coefficient

Example 1●●

For which values of $m$ does $x^2 + mx + 9 = 0$ have real roots?

$\Delta = m^2 - 36$

$m^2 - 36 \ge 0$, so $(m - 6)(m + 6) \ge 0$.

Real roots: Δ ≥ 0.

$m \le -6$ or $m \ge 6$

A concave-up quadratic in m, non-negative outside its zeroes.

Tangent lines

Substituting a line into a parabola gives a quadratic equation. Its discriminant tells you how many times they meet: two points ($\Delta > 0$), a tangent ($\Delta = 0$), or no meeting ($\Delta < 0$).

When is y = 2x + b a tangent to y = x²?

Slide b until Δ = 0. That is the tangent.

Example 2●●●

Find $k$ so that the line $y = kx - 4$ is tangent to $y = x^2$.

$x^2 = kx - 4$, so $x^2 - kx + 4 = 0$.

Tangent: $\Delta = k^2 - 16 = 0$.

A tangent meets the curve exactly once.

$k = 4$ or $k = -4$. There are two tangents from $(0, -4)$.

Flashcards

Real roots

$\Delta \ge 0$

Two distinct real roots

$\Delta > 0$

Rational roots

$\Delta$ is a perfect square.

Line is a tangent

Substitute, then set $\Delta = 0$.

Practice

$x^2 - 6x + k = 0$ has no real roots when:
$2x^2 - 7x + 3 = 0$ has $\Delta = 25$. Its roots are:
Find $c$ so that $y = 2x + c$ is tangent to $y = x^2 + 2$.
$x^2 + 4x + k = 0$ has real roots when:
Find the discriminant of $2x^2 + 3x - 2$.
The roots of $2x^2 + 3x - 2 = 0$ are:
Find the positive value of $m$ for which $y = mx$ is a tangent to $y = x^2 + 4$.
How many times does $y = 3x - 2$ meet $y = x^2$?