Q12 Interpret real and non-real quadratic roots using graphs

Ontario Grade 10 Mathematics

Quadratic Relations

What the parabola tells you about the solutions of a quadratic equation

Every quadratic equation of the form

ax^2 + bx + c = 0

has roots — values of

x

that make the equation true. In Grade 9 you learned that the graph of a quadratic relation is a U-shaped curve called a parabola. This lesson connects those two ideas: the roots of the equation are exactly the x-coordinates where the parabola crosses or touches the x-axis. Sometimes the parabola crosses the axis at two points, sometimes it just grazes the axis at one point, and sometimes it floats entirely above or below the axis without touching it at all. Each situation produces a different kind of root, and being able to read that situation from the graph — or to predict the graph from the equation — is the core skill of this lesson.

What you will learn

Explain what a root of a quadratic equation means in terms of the parabola's graph.
Identify whether a quadratic equation has two distinct real roots, one repeated real root, or no real roots by looking at how its parabola meets the x-axis.
Connect the number of x-intercepts of a parabola to the nature of the roots without solving the equation.
Use the discriminant to confirm what the graph already shows, and interpret a negative discriminant as the reason a parabola has no x-intercepts.
Describe non-real roots in plain language as values that cannot be plotted on a number line, and explain why they appear when the parabola misses the x-axis entirely.

Roots, Zeros, and the x-axis — a Quick Bridge from Grade 9

In Grade 9 you graphed quadratic relations such as

y = x^2 - 4

. You found the x-intercepts by setting

y = 0

and solving. Those x-intercepts are the same things as the roots of the equation

x^2 - 4 = 0

. The words root, zero, and x-intercept all point to the same idea: a value of

x

where the parabola sits exactly on the x-axis.

When the parabola meets the x-axis, the height of the curve (

y

) equals zero at that point. Because the parabola and the x-axis meet there, the x-coordinate of that meeting point satisfies the equation

ax^2 + bx + c = 0

. That is why solving the equation and finding x-intercepts are really the same calculation done two different ways.

Keep this connection in mind throughout the lesson: every conclusion you draw from the graph has a matching algebraic meaning, and every algebra result has a matching picture on the graph.

A root of $ax^2 + bx + c = 0$ is an x-value that makes the equation true.
Graphically, roots are the x-intercepts of the parabola $y = ax^2 + bx + c$ .
The words root, zero, and x-intercept describe the same point from different angles.

Three Possible Relationships Between a Parabola and the x-axis

A parabola can relate to the x-axis in exactly three ways, and each way tells you something different about the roots of the matching equation.

Case 1 — Two x-intercepts: The parabola crosses the x-axis at two separate points. This means the equation has two distinct real roots. The word real simply means the roots are ordinary numbers you can place on a number line. For example, if the parabola crosses at

x = -1

and

x = 3

, both

-1

and

3

are real numbers that satisfy the equation.

Case 2 — One x-intercept: The parabola just touches the x-axis at its vertex and does not cross through it. This means the equation has exactly one real root, sometimes called a repeated root or a double root. Both roots of the quadratic happen to be the same number. For example,

y = (x - 2)^2

has its vertex on the x-axis at

x = 2

, so the only root is

x = 2

.

Case 3 — No x-intercepts: The parabola floats entirely above the x-axis (opening upward) or entirely below it (opening downward). It never touches

y = 0

. This means the equation has no real roots. The solutions still exist mathematically, but they are non-real — they cannot be placed on a regular number line. At Grade 10, the key point is simply that these solutions exist but are not real numbers, which is why the parabola never reaches

y = 0

.

Two x-intercepts → two distinct real roots.
One x-intercept (vertex on axis) → one repeated real root.
No x-intercepts → no real roots; the roots are non-real.
Non-real roots cannot be plotted on a number line, which is why the parabola never crosses the x-axis.

The Discriminant: Algebra That Matches the Graph

The quadratic formula gives the roots of

ax^2 + bx + c = 0

as

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

. The expression under the square-root sign,

b^2 - 4ac

, is called the discriminant. It is usually written with the letter

D

or the symbol

\Delta

. You do not need to finish solving the equation to use the discriminant — you just evaluate it and check its sign.

If

b^2 - 4ac > 0

, the square root is a positive real number, so the

\pm

produces two different answers. The equation has two distinct real roots, matching Case 1 above. If

b^2 - 4ac = 0

, the square root equals zero, and both

+

and

-

give the same answer:

x = \frac{-b}{2a}

. That is the repeated root of Case 2. If

b^2 - 4ac < 0

, you are asked to take the square root of a negative number. No real number has a negative square, so there are no real roots — the roots are non-real, matching Case 3.

The discriminant is a fast tool: compute one expression, check whether it is positive, zero, or negative, and you immediately know the nature of the roots and the shape of the graph — all without drawing anything or finishing the full quadratic formula.

D = b^2 - 4ac

Discriminant: $D = b^2 - 4ac$ .
$D > 0$ : two distinct real roots, parabola crosses x-axis twice.
$D = 0$ : one repeated real root, vertex sits on the x-axis.
$D < 0$ : no real roots (roots are non-real), parabola does not touch the x-axis.
The discriminant connects the algebra directly to the graph without completing the full solution.

Reading Root Nature Directly from a Graph

Sometimes you are given a graph and asked to describe the roots, without any equation provided. The strategy is straightforward: count how many times the parabola crosses or touches the x-axis. Two crossing points mean two distinct real roots. One touching point (vertex on the axis) means one repeated real root. No crossing or touching at all means the roots are non-real.

You can also say something about approximate values when roots are real. If the parabola clearly crosses the x-axis between

x = 1

and

x = 2

, the root in that interval is a real number somewhere in that range. When there are no x-intercepts, you cannot read off root values from the graph — you can only confirm that the roots are non-real.

An important reasoning step: if the parabola opens upward and its vertex is above the x-axis, it will never come down to cross the x-axis, so the roots must be non-real. If it opens upward and the vertex is below the x-axis, it must cross upward through the x-axis on both sides, giving two real roots. This vertex-and-direction reasoning lets you interpret root nature at a glance.

Count x-intercepts on the graph to determine root nature instantly.
A parabola opening upward with vertex above the x-axis has non-real roots.
A parabola opening upward with vertex below the x-axis has two distinct real roots.
Approximate real root values can be read from the graph; non-real roots cannot.

Putting It All Together: Graph, Equation, and Root Nature

The graph and the discriminant are two lenses on the same situation. The graph shows you the geometry; the discriminant gives you the algebra. They must always agree: a positive discriminant always goes with a parabola that has two x-intercepts, a zero discriminant always goes with a vertex exactly on the x-axis, and a negative discriminant always goes with a parabola that floats away from the x-axis entirely.

When you are given an equation, compute

D = b^2 - 4ac

first. Its sign tells you what to expect on the graph. When you are given a graph, count the x-intercepts first. That count tells you the sign of the discriminant — even if you never calculate it. Practising both directions — algebra to graph and graph to algebra — builds the flexible understanding that this topic requires.

Remember that non-real roots are not errors or mistakes. They are a legitimate answer: the equation has solutions, but those solutions are not real numbers. Saying 'no real roots' is a complete and correct mathematical statement at this level.

The graph and the discriminant always give the same conclusion about root nature.
Practice interpreting in both directions: given a graph, state the root nature; given an equation, use D to predict the graph.
'No real roots' is a complete, correct answer — non-real roots exist but cannot be shown on a standard number-line graph.

Summary: Three Cases for Quadratic Roots and Their Graphs

Discriminant D = b² − 4ac	Number of x-intercepts	Type of Roots	Graph Description
D > 0	2	Two distinct real roots	Parabola crosses x-axis at two separate points
D = 0	1	One repeated real root	Vertex of parabola sits exactly on the x-axis
D < 0	0	No real roots (non-real roots)	Parabola floats entirely above or below the x-axis

Worked example

Example 1 — Using the Discriminant to Predict and Interpret the Graph

Consider the quadratic equation

2x^2 - 4x + 5 = 0

. Without solving the equation fully, determine the nature of its roots and describe what the graph of

y = 2x^2 - 4x + 5

looks like relative to the x-axis.

Identify a, b, and c
Write the equation in standard form $ax^2 + bx + c = 0$ and read off the three coefficients.
$a = 2, b = -4, c = 5$
Write the discriminant formula
The discriminant is $D = b^2 - 4ac$ . Substituting the values gives a single number that determines root nature.
$D = (-4)^2 - 4(2)(5)$
Calculate D
Evaluate $(-4)^2 = 16$ and $4 \times 2 \times 5 = 40$ , then subtract.
$D = 16 - 40 = -24$
Interpret the sign of D
Because $D = -24 < 0$ , the equation has no real roots. The roots are non-real, meaning they cannot be placed on a number line.
$D < 0 \implies \text{no real roots}$
Describe the graph
Since $a = 2 > 0$ , the parabola opens upward. Because there are no real roots, the parabola never crosses or touches the x-axis. It sits entirely above the x-axis.

Answer:

D = -24 < 0

, so the equation has no real roots. The graph of

y = 2x^2 - 4x + 5

is a parabola opening upward that lies entirely above the x-axis with no x-intercepts.

Check: Quick reasonableness check: find the vertex x-coordinate using

x = -\frac{b}{2a} = -\frac{-4}{2(2)} = 1

. The vertex y-value is

y = 2(1)^2 - 4(1) + 5 = 2 - 4 + 5 = 3

. The vertex is at

(1, 3)

, which is above the x-axis. With the parabola opening upward, it never comes down to the x-axis. This confirms no x-intercepts and no real roots.

Worked example

Example 2 — Reading Root Nature from a Graph Description and Verifying Algebraically

A student sketches the parabola

y = -x^2 + 6x - 9

and claims it touches the x-axis at exactly one point. (a) Use the discriminant to confirm or correct the claim. (b) Find the repeated root and identify the point on the graph.

Identify a, b, and c
The equation is already in standard form. Read off the coefficients carefully, including the negative sign on $x^2$ .
$a = -1, b = 6, c = -9$
Calculate the discriminant
Substitute into $D = b^2 - 4ac$ . Pay careful attention to the signs when multiplying.
$D = 6^2 - 4(-1)(-9)$
Evaluate each part
Compute $6^2 = 36$ . Then compute $4 \times (-1) \ times (-9)$ : two negatives multiply to give a positive, so $4(-1)(-9) = 36$ .
$D = 36 - 36 = 0$
Interpret D = 0
Because $D = 0$ , the equation has exactly one real root — a repeated root. The student's claim is confirmed: the parabola touches the x-axis at exactly one point.
$D = 0 \implies \text{one repeated real root}$
Find the repeated root
When $D = 0$ , the quadratic formula simplifies to $x = \frac{-b}{2a}$ . Substitute the values to find the single root.
$x = \frac{-6}{2(-1)} = \frac{-6}{-2} = 3$
Identify the graph point
The repeated root $x = 3$ is the x-coordinate of the vertex, and the parabola touches the x-axis there. The point on the graph is $(3, 0)$ . Since $a = -1 < 0$ , the parabola opens downward and sits below the x-axis everywhere except at this one touching point.
$(3,\ 0)$

Answer:

D = 0

, confirming one repeated real root at

x = 3

. The graph of

y = -x^2 + 6x - 9

is a downward-opening parabola that touches the x-axis at exactly one point,

(3, 0)

.

Check: Verify by substituting

x = 3

into the original equation:

-(3)^2 + 6(3) - 9 = -9 + 18 - 9 = 0

. ✓ The point

(3, 0)

lies on the parabola, confirming it is both the vertex and the only x-intercept.

Common mistakes and how to avoid them

Thinking that 'no real roots' means the equation has no solutions at all.

Correction: The equation still has solutions — they just are not real numbers. At Grade 10, the correct statement is 'the roots are non-real,' not 'there are no roots.'

Forgetting that

b^2

is always positive (or zero), but

4ac

can be positive or negative depending on the signs of

a

and

c

. Students sometimes drop a negative sign when computing

4ac

.

Correction: Write out each multiplication step carefully. For example, if

a = -1

and

c = -9

, then

4ac = 4(-1)(-9) = +36

, not

-36

.

Concluding 'two real roots' just because the parabola is wide or steep, without checking x-intercepts.

Correction: The number of real roots depends entirely on the number of x-intercepts, not on the width or steepness of the parabola.

Saying a parabola that opens downward and has no x-intercepts has 'two real roots' because it looks like it should cross the axis.

Correction: A downward-opening parabola with its vertex below the x-axis has no x-intercepts and no real roots. Check the vertex position and the opening direction together.

Mixing up the repeated-root case: thinking one x-intercept means only one root exists and the equation is not truly quadratic.

Correction: A quadratic always has exactly two roots (counting repetition). When

D = 0

, both roots happen to be the same value — that is why we say 'one repeated root,' not 'one root total.'

Lesson summary

A root of a quadratic equation is an x-value that satisfies $ax^2 + bx + c = 0$ ; graphically, roots are the x-intercepts of the parabola $y = ax^2 + bx + c$ .
A parabola can cross the x-axis at two points (two distinct real roots), touch it at one point (one repeated real root), or miss it entirely (no real roots — the roots are non-real).
The discriminant $D = b^2 - 4ac$ predicts which case applies: $D > 0$ gives two real roots, $D = 0$ gives one repeated real root, and $D < 0$ gives no real roots.
Non-real roots are not errors; they are valid solutions that cannot be expressed as ordinary numbers on a number line, which is why the parabola never reaches the x-axis in that case.
The graph and the discriminant always agree: use whichever is given first, then confirm with the other.
Checking the vertex position and the direction the parabola opens is a reliable way to predict root nature directly from the equation without computing the discriminant.

Check your understanding

Question 1

The parabola

y = x^2 + 2x + 10

opens upward and its vertex is above the x-axis. What can you conclude about the roots of

x^2 + 2x + 10 = 0

?

Two distinct real roots, because all upward parabolas cross the x-axis twice.
One repeated real root, because the vertex is the only point that matters.
No real roots, because the parabola never touches the x-axis.
Two real roots equal in value, because the vertex is above the axis.

Show answer and explanation

No real roots, because the parabola never touches the x-axis.

An upward-opening parabola whose vertex is above the x-axis can never come down to meet the x-axis. There are no x-intercepts, so the equation has no real roots. You can confirm this with the discriminant:

D = 2^2 - 4(1)(10) = 4 - 40 = -36 < 0

.

Question 2

Which value of

D = b^2 - 4ac

corresponds to a parabola that touches the x-axis at exactly one point?

$D = 9$
$D = -1$
$D = 0$
$D = 4$

Show answer and explanation

D = 0

When

D = 0

, the quadratic formula produces only one value:

x = \frac{-b}{2a}

. This is the repeated root, and it corresponds to the vertex sitting exactly on the x-axis — one touching point and no crossing.

Question 3

A student calculates

D = b^2 - 4ac = 25

for a quadratic equation. What does this tell you about the graph of the matching parabola?

The parabola does not intersect the x-axis.
The parabola touches the x-axis at one point only.
The parabola crosses the x-axis at two distinct points.
The parabola's vertex is on the x-axis.

Show answer and explanation

The parabola crosses the x-axis at two distinct points.

Because

D = 25 > 0

, the square root

\sqrt{25} = 5

is a positive real number. The

\pm

in the quadratic formula then produces two different x-values, meaning two distinct real roots and two x-intercepts on the graph.

Question 4

The equation

3x^2 - 6x + 3 = 0

is given. A classmate says the graph has two x-intercepts. Are they correct?

Yes, because the coefficient of $x^2$ is positive.
No, the graph has no x-intercepts because the roots are non-real.
No, the graph touches the x-axis at exactly one point because $D = 0$ .
Yes, because every quadratic graph must cross the x-axis at least once.

Show answer and explanation

No, the graph touches the x-axis at exactly one point because

D = 0

.

Calculate

D = (-6)^2 - 4(3)(3) = 36 - 36 = 0

. Because

D = 0

, there is exactly one repeated real root, meaning the vertex sits on the x-axis at one point only — not two. The classmate is incorrect.

Key terms

Root (of a quadratic equation): A value of $x$ that makes the equation $ax^2 + bx + c = 0$ true. Also called a zero of the related function.
x-intercept: A point where a graph crosses or touches the x-axis, written as $(x, 0)$ . The x-coordinate of an x-intercept is a root of the matching equation.
Parabola: The U-shaped (or ∩-shaped) curve that is the graph of any quadratic relation $y = ax^2 + bx + c$ .
Discriminant: The expression $D = b^2 - 4ac$ found under the square-root sign in the quadratic formula. Its sign tells you the number and type of roots without completing the full solution.
Real root: A root that is an ordinary number that can be placed on a number line, such as $3$ , $-\frac{1}{2}$ , or $\sqrt{5}$ .
Non-real root: A root that cannot be expressed as a point on a number line. Non-real roots appear when $D < 0$ , because you would need to take the square root of a negative number.
Repeated root: The single root that appears when $D = 0$ . Both roots of the quadratic are equal to the same value, so we say the root is repeated or double.
Vertex: The turning point of a parabola — its highest point if it opens downward, or its lowest point if it opens upward. When $D = 0$ , the vertex lies exactly on the x-axis.

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About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 10 Mathematics (MPM2D), study topic Q12. It is a study resource, not an official curriculum publication.

Official curriculum reference

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Q12 · Interpret real and non-real quadratic roots using graphs

Ontario Grade 10 Mathematics

What the parabola tells you about the solutions of a quadratic equation

What you will learn

Roots, Zeros, and the x-axis — a Quick Bridge from Grade 9

Three Possible Relationships Between a Parabola and the x-axis

The Discriminant: Algebra That Matches the Graph

Reading Root Nature Directly from a Graph

Putting It All Together: Graph, Equation, and Root Nature

Summary: Three Cases for Quadratic Roots and Their Graphs

Example 1 — Using the Discriminant to Predict and Interpret the Graph

Example 2 — Reading Root Nature from a Graph Description and Verifying Algebraically

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Question 4

Key terms

Continue through MPM2D

About this lesson