Quadratic Relations of the Form y = ax² + bx + c · Quadratic Expressions and Equations

Real and Non-Real Roots: The Discriminant

3.4 4.2About 65 minutes

Learning goals

I can explain, using graphs, why a quadratic equation can have two, one or no real roots.
I can use the discriminant b² − 4ac to determine the number of real roots without solving.
I can relate the number of real roots to the number of x-intercepts of the graph.
I can use the discriminant to answer questions such as "Does the ball ever reach this height?"

Words to know

discriminant
The expression b² − 4ac from the quadratic formula. Its sign tells how many real roots ax² + bx + c = 0 has.

real number
Any number on the number line, including integers, fractions and decimals such as √2.

non-real roots
Roots that are not real numbers. They occur when the discriminant is negative; the graph has no x-intercepts.

double root
One root that occurs twice, when the discriminant is 0. The vertex is on the x-axis.

distinct
Different from each other.

Three parabolas, three stories

Graph these three relations on the same axes (with technology, or with tables of values):

y = x² − 4x + 3
y = x² − 4x + 4
y = x² − 4x + 5

They differ only in the constant term, so each is a vertical translation of the others. Count the x-intercepts: the first crosses the x-axis twice, the second touches it once, and the third never reaches it. So the equations x² − 4x + c = 0 have two, one and no real roots. How could you tell without graphing?

y = x² − 4x + c for c = 3 (solid), c = 4 (dashed) and c = 5 (dotted). Raising c slides the parabola up: two x-intercepts (at 1 and 3), then one (at 2), then none.

What the quadratic formula tells us

Use the formula on each equation and watch the number under the square root:

x² − 4x + 3 = 0: b² − 4ac = 16 − 12 = 4. x = (4 ± 2) ÷ 2, so x = 3 or x = 1. Two roots.
x² − 4x + 4 = 0: b² − 4ac = 16 − 16 = 0. x = (4 ± 0) ÷ 2 = 2. One root.
x² − 4x + 5 = 0: b² − 4ac = 16 − 20 = −4. x = (4 ± √(−4)) ÷ 2. No real number squares to −4, so there are no real roots.

The expression b² − 4ac decides everything. It is called the discriminant, because it discriminates (tells apart) the three cases.

The three cases of the discriminant. The dots show the x-intercepts, which are the real roots of ax² + bx + c = 0.

Real and non-real roots

When b² − 4ac < 0, the formula asks for the square root of a negative number. That is not a real number (the numbers on the number line), so we say the equation has no real roots, or that its roots are non-real. Mathematicians do have numbers that make sense of √(−4), but they belong to later courses. In this course, a negative discriminant means: the graph has no x-intercepts.

A bonus fact: if the discriminant is a perfect square (0, 1, 4, 9, 16, ...) and a, b, c are integers, the roots are rational and the expression can be factored over the integers.

Discriminant b² − 4ac	Number of real roots	Graph of y = ax² + bx + c
positive	two different real roots	crosses the x-axis twice
zero	one real root (a double root)	vertex touches the x-axis
negative	no real roots (non-real roots)	does not meet the x-axis

Worked example

Example 1: How many roots?

Without solving, determine the number of real roots of (a) 2x² − 3x + 5 = 0, (b) 9x² − 12x + 4 = 0, (c) x² + 3x − 7 = 0.

(a) a = 2, b = −3, c = 5: b² − 4ac = 9 − 40 = −31 < 0. No real roots.
(b) a = 9, b = −12, c = 4: 144 − 144 = 0. One real root. (Indeed 9x² − 12x + 4 = (3x − 2)².)
(c) a = 1, b = 3, c = −7: 9 + 28 = 37 > 0. Two real roots.

(a) none, (b) one, (c) two.

Worked example

Example 2: Use the vertex form

How many zeros does y = −2(x − 1)² + 8 have?

The vertex is (1, 8), which is above the x-axis.
a = −2 < 0, so the parabola opens down, towards the x-axis.
So it must cross the x-axis twice. (Check: (x − 1)² = 4 gives x = 3 or x = −1.)

Two zeros. Rule of thumb: if the vertex is on one side of the x-axis and the parabola opens towards the axis, there are two zeros; if it opens away, there are none; if the vertex is on the axis, there is one.

Worked example

Example 3: Find k

For what values of k does x² + kx + 9 = 0 have exactly one real root?

One root means b² − 4ac = 0.
k² − 4(1)(9) = 0, so k² = 36.
k = 6 or k = −6.
Check: x² + 6x + 9 = (x + 3)² and x² − 6x + 9 = (x − 3)². Each has one root ✓.

k = 6 or k = −6

Worked example

Example 4: Can the ball get that high?

A ball's height is h = −4.9t² + 15t + 1 (metres, seconds). Can the ball reach 20 m?

Set h = 20: −4.9t² + 15t + 1 = 20, so −4.9t² + 15t − 19 = 0.
Discriminant: 15² − 4(−4.9)(−19) = 225 − 372.4 = −147.4.
It is negative, so there is no time when h = 20.
(The maximum height is 1 + 225 ÷ 19.6 ≈ 12.5 m, found by completing the square or with technology.)

No, the ball never reaches 20 m.

Watch out

Write the equation as ax² + bx + c = 0 before reading a, b and c.
Use brackets for negatives: (−5)² = 25, and −4(−4.9)(−19) is negative.
"No real roots" does not mean "no answer": it is the answer. Say what it means in context (for example, the ball never reaches that height).

Hands-on activity

Slide the parabola

You need: a sheet of thin paper (tracing paper or printer paper you can see through), graph paper, a pencil and a ruler.

On graph paper, draw x- and y-axes and plot y = x² carefully using x = −3 to 3. Go over the curve darkly.
Lay the thin paper on top and trace the parabola and a short piece of the y-axis (so you can keep it lined up).
Slide the thin paper straight up 2 squares, then straight down 2 squares, keeping the traced y-axis on the real y-axis. Each time, count how many times the traced curve meets the x-axis.
Write the equation for each position: y = x² + 2 and y = x² − 2. Work out the discriminant of each.
Flip the thin paper over (so the parabola opens down) and repeat.

Model answer: y = x² + 2 has discriminant 0 − 8 = −8 and no x-intercepts; y = x² has discriminant 0 and one; y = x² − 2 has discriminant 8 and two. Flipped, y = −x² + 2 has discriminant 8 and two x-intercepts.

Just for fun

Why did the parabola with a negative discriminant never finish its road trip?

It could see the x-axis the whole time, but it never had any real way to get there.

Your turn

Find the discriminant of x² + 6x + 9 = 0 and state the number of real roots.

Find the discriminant of x² − 5x + 2 = 0 and state the number of real roots.

Find the discriminant of 3x² + 2x + 1 = 0 and state the number of real roots.

How many x-intercepts does y = 4x² − 12x + 9 have?

Without expanding, how many zeros does y = −2(x − 1)² + 8 have? Then find them.

How many zeros does y = 3(x + 2)² + 5 have? Explain.

Find k so that x² + kx + 16 = 0 has exactly one real root.

More practice: warm-up

Write the formula for the discriminant.

If the discriminant is positive, how many real roots are there?

Find the discriminant of x² − 2x − 3 = 0.

Find the discriminant of x² + x + 1 = 0. How many real roots?

A parabola's vertex is on the x-axis. What is the discriminant of its equation?

More practice: on your own

Find the discriminant of 2x² − 7x + 6 = 0. What does its value tell you about factoring?

How many real roots does 0.5x² + 3x + 4.5 = 0 have? Find them.

For what values of k does x² + 6x + k = 0 have two real roots?

For what values of k does 2x² + kx + 8 = 0 have no real roots?

A ball's height is h = −4.9t² + 15t + 1. Use the discriminant to decide whether it reaches 20 m.

Challenge

Does the ball in h = −4.9t² + 15t + 1 reach 12 m? If so, when is it at 12 m (two decimal places)?

Show that x² + bx − 1 = 0 has two real roots for every value of b.

The line y = 2x + k and the parabola y = x² meet at exactly one point. Find k and the point.

Check your understanding

Quiz

What is the discriminant of 2x² + 3x − 1 = 0?

If b² − 4ac < 0, the graph of y = ax² + bx + c:

Which equation has exactly one real root?

How many zeros does y = 2(x − 3)² + 4 have?

For which values of k does x² + kx + 25 = 0 have a double root?

The vertex of a parabola lies exactly on the x-axis. What is true about b² − 4ac?

For parents and teachers

Big idea: Expectation 3.4 asks students to interpret real and non-real roots of quadratic equations, through investigation using graphing technology, and relate the roots to the x-intercepts. The discriminant b² − 4ac is the algebraic tool that predicts two, one or no real roots without solving.

How to help: The tracing-paper activity makes the idea very visual: as a parabola slides, it meets the x-axis twice, once, or not at all. Ask your teen to predict the number of x-intercepts from the vertex and direction of opening, then confirm with the discriminant.

Common difficulties:

Sign errors in −4ac when a or c is negative.
Thinking "no real roots" means the question is wrong; it is a valid answer.
Forgetting to rearrange to = 0 before reading a, b and c (Example 4).