Functions and Quadratic Equations

Try this

Click the “Function” card and drag $x$ . When does the point on the graph land exactly on the $x$ -axis? What is $y$ then?
Click the “Equation” card. The two roots of the equation are the same thing as which two points in the picture?
Drag $c$ upward one notch at a time. How do the two crossing points move? When do they merge into one? And after that? Watch the value of $Δ$ at the same time.
Try several values of $b$ and $c$ . Exactly halfway between the two roots there is always which line in the picture?

The same expression $a x^{2} + b x + c$ gets a different name depending on the question:

“As $x$ changes, how does $y$ follow?” It is a function.
“Which $x$ make it equal $0$ ?” It is an equation.
“Which $x$ make it greater than $0$ ?” It is an inequality (next stop).

Predict first

The parabola $y = x^{2} - 4 x + c$ meets the $x$ -axis at exactly one point. What is $c$ ?

Choose an answer first; then you can see the result.

Why: the discriminant measures the height of the vertex

y = x² − 2x + c: as c changes, so does the number of crossings

Step 1 / 5

Complete the square first
$y = x^{2} - 2 x + c = (x - 1)^{2} + (c - 1)$ . The vertex is $(1, c - 1)$ , and the parabola opens upward.

$y = (x - 1)^{2} + (c - 1)$
Vertex below the x-axis
Vertex exactly on the x-axis
Vertex above the x-axis
The general rule

Where the roots are: locating roots

Sometimes the question is not just “are there roots?” but “where are they?”. For example: both roots of the equation are greater than $1$ . Solving for the roots directly is messy here, but the picture is easy: put the parabola where it has to be and see what that requires.

Drag

b

c

k

on the

x

-axis. Each target needs different conditions. Guess which ones before you look.

For a parabola that opens upward, the five common targets need these conditions:

Target	Conditions
Both roots greater than $k$	$Δ \geq 0$ , axis to the right of $k$ , $f (k) > 0$
Both roots less than $k$	$Δ \geq 0$ , axis to the left of $k$ , $f (k) > 0$
One root less than $k$ , one greater	$f (k) < 0$ (that is all!)
Both roots between $m$ and $n$	$Δ \geq 0$ , axis between $m$ and $n$ , $f (m) > 0$ , $f (n) > 0$
Exactly one root between $m$ and $n$	$f (m) \cdot f (n) < 0$

No need to memorize the table. Each condition just describes one thing in the picture: the graph reaches the $x$ -axis, it sits to the left or right overall, a given point is outside or between the roots.

Example: both roots of x² − 2mx + m + 2 = 0 are greater than 1. Find the range of m.

Step 1 / 5

Try m = 0 to start with
The equation is $x^{2} + 2 = 0$ . The graph does not even reach the $x$ -axis: there are no roots at all. First condition: $Δ \geq 0$ .

$Δ = 4 m^{2} - 4 (m + 2) \geq 0$
m = −2: roots, but on the left
m = 4: the axis is far enough right, but…
m = 2.5: all three lights on
Combine the three conditions

One problem, several solutions

For which $k$ does the equation $x^{2} - 4 x + k = 0$ have one root greater than $3$ and the other less than $3$ ?

From the picture: f(3) < 0

The parabola opens upward. $3$ lying between the roots is the same as the graph being below the $x$ -axis at $x = 3$ :

f (3) = 9 - 12 + k < 0, k < 3.

A graph that opens upward and has a point below the $x$ -axis must cross the $x$ -axis twice, so $Δ > 0$ holds automatically.

CostDone in one step; you have to see why Δ needs no extra check.

Vieta's formulas

Call the roots $x_{1}$ and $x_{2}$ . We need $(x_{1} - 3) (x_{2} - 3) < 0$ . Expand and use $x_{1} + x_{2} = 4$ , $x_{1} x_{2} = k$ :

x_{1} x_{2} - 3 (x_{1} + x_{2}) + 9 = k - 12 + 9 < 0, k < 3.

We also need two real roots: $Δ = 16 - 4 k > 0$ , so $k < 4$ . Together the two conditions still give $k < 3$ .

CostPure algebra, no picture; you must check Δ separately.

Remove one condition

Break it

Counterexample lab: remove one condition

Conclusion: If the axis is to the right of $k$ and $f (k) > 0$ , then both roots are greater than $k$

With all three conditions in use, “all conditions hold” and “both roots really are greater than $k$ ” are always true together or false together. Drag anything you like: the verdict box below never raises the alarm.

Practice

Exercise 1Basic

The parabola $y = x^{2} - 5 x + 6$ meets the $x$ -axis at two points. Give their $x$ -coordinates.

Exercise 2Basic

The parabola $y = x^{2} + 6 x + m$ meets the $x$ -axis at exactly one point. Find $m$ .

Exercise 3Advanced

The parabola $y = x^{2} - 2 x - 8$ cuts off a segment on the $x$ -axis. How long is that segment?

Exercise 4Advanced

The equation $x^{2} + (m - 3) x + m = 0$ has one root greater than $1$ and the other less than $1$ . What is the largest integer $m$ that works?

Exercise 5Challenge

The equation $x^{2} - 2 m x + m + 6 = 0$ has two real roots (possibly equal), and both are greater than $1$ . How many integers $m$ work?

Try this

Why: the discriminant measures the height of the vertex

y = x² − 2x + c: as c changes, so does the number of crossings

Where the roots are: locating roots

Example: both roots of x² − 2mx + m + 2 = 0 are greater than 1. Find the range of m.

From the picture: f(3) < 0

Vieta's formulas

Remove one condition

Counterexample lab: remove one condition

Practice

Mistake Museum · Exhibits from this stop

“Two real roots”, forgetting it must be a quadratic first

“Only one crossing”, thinking only of tangency