A. Quadratic Functions · Quadratic Expressions and Equations

Real Roots, x-Intercepts and the Discriminant

A1.7About 65 minutes

Learning goals

I can explain that the real roots of a quadratic equation are the x-intercepts of its graph.
I can use the discriminant to tell whether an equation has two, one or no real roots.
I can use the discriminant to answer questions in context and to find unknown coefficients.

Words to know

real root
A real-number solution of an equation; for a quadratic, an x-intercept of the graph.

discriminant
D = b² − 4ac for ax² + bx + c = 0. Its sign gives the number of real roots.

double root
The single repeated root when D = 0; the graph's vertex is on the x-axis.

no real roots
When D < 0: no real number satisfies the equation, and the graph has no x-intercepts.

rational / irrational
A rational number can be written as a fraction of integers; an irrational one, like √2, cannot.

How many roots?

Before you spend time solving an equation, it helps to know how many real roots to expect. Some quadratic equations have two, some have one, and some have none at all. In the lesson Factors and x-Intercepts you saw graphs that cross the x-axis twice, touch it once, or miss it completely. In this lesson you will predict which case you are in without solving, using one number: the discriminant.

Real roots are x-intercepts

The real roots of ax² + bx + c = 0 are the x-coordinates of the points where the graph of y = ax² + bx + c meets the x-axis. So:

two x-intercepts ↔ two real roots
the vertex on the x-axis (one x-intercept) ↔ one real root (a double root)
no x-intercepts ↔ no real roots

Try it with three related parabolas that differ only in their constant term.

Three parabolas: y = x² − 2x − 3 (thick) crosses the x-axis at −1 and 3; y = x² − 2x + 1 (dashed) touches it at 1; y = x² − 2x + 3 (thin) never reaches it.

The discriminant decides

In the quadratic formula x = (−b ± √(b² − 4ac)) / (2a), everything depends on the number under the root, D = b² − 4ac:

If D > 0, √D is a positive real number. Adding it and subtracting it give two different real roots.
If D = 0, ±√0 adds nothing, so both roots are −b/(2a): one real root.
If D < 0, √D is not a real number, so there are no real roots and the graph has no x-intercepts.

For the three parabolas above: D = 4 + 12 = 16, D = 4 − 4 = 0, and D = 4 − 12 = −8. That matches the graphs exactly.

The sign of the discriminant tells you how many times the parabola meets the x-axis. These are sketches with a > 0; the same rules hold when a < 0 and the parabola opens down.

Worked example

Example 1: Count the roots without solving

How many real roots does each equation have? (a) 2x² − 3x + 5 = 0 (b) 4x² − 12x + 9 = 0 (c) 3x² + x − 7 = 0

(a) D = (−3)² − 4(2)(5) = 9 − 40 = −31 < 0: no real roots.
(b) D = (−12)² − 4(4)(9) = 144 − 144 = 0: one real root. (In fact 4x² − 12x + 9 = (2x − 3)², so x = 1.5.)
(c) D = 1² − 4(3)(−7) = 1 + 84 = 85 > 0: two real roots. Since 85 is not a perfect square, the roots are irrational and the trinomial does not factor over the integers.

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

Bonus: does it factor?

If a, b and c are integers and D is a perfect square (0, 1, 4, 9, 16, 25, ...), the roots are rational and ax² + bx + c factors over the integers. If D is positive but not a perfect square, the roots are irrational: use the formula. This is a quick way to decide which method to use.

Worked example

Example 2: Will the ball get that high?

A ball's height is h = −4.9t² + 20t + 1.5 (metres, seconds), the ball from Using the Quadratic Formula. Does it ever reach 25 m?

Set h = 25: −4.9t² + 20t + 1.5 = 25, so −4.9t² + 20t − 23.5 = 0.
D = 20² − 4(−4.9)(−23.5) = 400 − 460.6 = −60.6.
D < 0, so the equation has no real roots: the graph never reaches the line h = 25.
Interpretation: the ball never reaches 25 m. (Its maximum height is about 21.9 m.)

No. The discriminant is negative, so the height 25 m is never reached.

Worked example

Example 3: Finding an unknown coefficient

(a) For what values of k does x² + kx + 9 = 0 have exactly one real root? (b) For what values of k does 2x² − 4x + k = 0 have two real roots?

(a) One root means D = 0: k² − 4(1)(9) = 0, so k² = 36 and k = 6 or k = −6. Check: x² + 6x + 9 = (x + 3)² ✓.
(b) Two roots means D > 0: (−4)² − 4(2)(k) > 0, so 16 − 8k > 0.
Then 16 > 8k, so k < 2. For example, k = 0 gives 2x² − 4x = 0, with roots 0 and 2 ✓.

(a) k = ±6 (b) k < 2

Just for fun

The discriminant is the bouncer of quadratic equations. It checks b² − 4ac at the door: negative? Sorry, no real roots tonight.

Hands-on activity

Discriminant detective with a string

You need: a piece of string or a long ruler, and graph paper.

On graph paper, draw the parabola y = x² carefully (plot x = −3 to 3).
Hold the string horizontally at the heights y = 4, y = 0 and y = −2. Count how many times it meets the parabola each time.
Meeting the line y = k means solving x² = k, or x² − k = 0, where a = 1, b = 0 and c = −k. Compute the discriminant 0² − 4(1)(−k) = 4k for each height.
Write a sentence connecting the sign of 4k with the number of meeting points.

What to look for: at y = 4, D = 16 > 0 and the string meets the curve twice (x = ±2). At y = 0, D = 0 and it touches once (the vertex). At y = −2, D = −8 < 0 and it misses. Positive D: two; zero: one; negative: none.

Your turn

Find the discriminant of x² + 5x + 3 = 0 and state the number of real roots.

Find the discriminant of 2x² + 4x + 5 = 0 and state the number of real roots.

Find the discriminant of 9x² − 6x + 1 = 0 and state the number of real roots.

How many x-intercepts does y = x² − 3x + 4 have?

How many x-intercepts does y = −2x² + 3x + 1 have?

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

A ball's height is h = −4.9t² + 14t + 1. Does it reach 12 m?

More practice: warm-up

Write the discriminant of ax² + bx + c = 0.

D = 20. How many real roots?

D = 0. What does the graph do at the x-axis?

D = −5. How many x-intercepts?

Find D for x² − 4x + 4 = 0.

More practice: on your own

Find D for 3x² − 2x − 8 = 0. How many roots? Does it factor?

Show that 0.5x² − 3x + 4.5 = 0 has exactly one root, and find it.

For what values of k does 2x² − 4x + k = 0 have no real roots?

For what values of k does kx² + 6x + 3 = 0 have no real roots? (Assume k ≠ 0.)

The canoe revenue is R = −10x² + 90x + 3600. Can the revenue ever reach $3900? Use the discriminant.

Challenge

Show that x² + bx − 5 = 0 has two real roots for every value of b.

The parabola y = x² + 6x + c touches the x-axis at exactly one point. Find c and the point.

A ball is thrown from 1.5 m with launch speed v m/s, so h = −4.9t² + vt + 1.5. What is the smallest v that lets it reach 10 m?

Check your understanding

Quiz

The discriminant of 2x² + 3x − 1 = 0 is:

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

Which equation has exactly one real root?

A height equation h = 30 gives D = −12. This means:

For what value of k does x² + kx + 25 = 0 have one root (with k > 0)?

Integers a, b, c give D = 49. What do you know?

For parents and teachers

Big idea (A1.7): the real roots of a quadratic equation are the x-intercepts of its graph, and the discriminant b² − 4ac tells how many there are (two, one or none) without solving.

How to help: ask "How many answers should you get?" before your teen solves any quadratic equation. Predicting with the discriminant, then checking, builds confidence.

Common difficulties:

Sign errors: −4ac becomes positive when a and c have opposite signs.
Thinking "no real roots" means the work is wrong. It is a legitimate answer, and in context it has a meaning (the ball never reaches that height).
Solving inequalities for k: dividing by a negative number reverses the inequality.
Confusing "one root" with "one x-intercept" on a graph that also has a y-intercept. Only x-intercepts count.