Why Does a Quadratic Equation Have Two Roots?

Students learn very early that a quadratic equation has two roots, and then spend the rest of the chapter finding them without ever asking why. The question is worth asking, because the answer ties together three ideas that the chapter treats separately: factorisation, the graph, and the discriminant. Once you see why there are two roots, you also see why sometimes there are fewer.

This guide explains the reason from three angles, uses a ball thrown in the air as a real example, and then looks at what happens when the number of roots is one or none. Before you start, you may want to skim this introduction to quadratic equations and the page on solving a quadratic equation by factorisation, and keep the chapter notes handy.

What a Root Is

A quadratic equation has the form ax² + bx + c = 0, where a, b and c are real numbers and a ≠ 0. A root is a value of x that makes the equation true. Roots are also called the zeroes of the polynomial ax² + bx + c, because they are the values of x where the polynomial equals zero. The relationship between the two names is also at the heart of polynomial division.

Reason One: Factorisation

If a quadratic can be factorised, it splits into two linear factors:

ax² + bx + c = a(x − α)(x − β)

The equation becomes a(x − α)(x − β) = 0. A product of numbers is zero only when at least one of them is zero. So either x − α = 0 or x − β = 0, giving x = α or x = β. Two linear factors, two roots.

Example. x² − 5x + 6 = 0 factorises as (x − 2)(x − 3) = 0, so x = 2 or x = 3.

Notice that this also explains why a quadratic cannot have three roots. It has only two linear factors to set to zero. In general, a polynomial of degree n can have at most n roots, a fact you also meet in the chapter on polynomials, where a cubic has at most three zeroes.

Reason Two: The Parabola

The graph of y = ax² + bx + c is a smooth U-shaped curve called a parabola. The roots are the x-values where the curve crosses the x-axis, because that is where y = 0.

A U-shaped curve can cross a straight line in at most two places. It might cut the x-axis twice, touch it once at the bottom of the U, or miss it entirely. That gives three cases, which match exactly what the discriminant tells you.

Reason Three: The Formula and the ±

Completing the square on ax² + bx + c = 0 leads to the quadratic formula:

x = (−b ± √(b² − 4ac)) ÷ 2a

The ± sign is where the two roots come from: one with the plus and one with the minus. The expression under the root, D = b² − 4ac, is the discriminant. It decides how many real roots there are, because you can only take the square root of a non-negative number.

So when D = 0, the equation still has “two” roots, but they are the same number, and we say the root is repeated. That is why the textbook says “two equal roots” and not “one root”.

Three Examples, One for Each Case

A Ball in the Air: Two Roots in Real Life

A ball is thrown straight up, and its height after t seconds is h = 20t − 5t² metres. When is it 15 m above the ground?

Set 20t − 5t² = 15, so 5t² − 20t + 15 = 0. Divide by 5: t² − 4t + 3 = 0, which factorises as (t − 1)(t − 3) = 0. So t = 1 second or t = 3 seconds.

Both answers are right, and both make physical sense. The ball passes 15 m on the way up at 1 second, and again on the way down at 3 seconds. That is the clearest way to feel why a quadratic has two roots: a thing that goes up and comes down passes through every height below its peak twice.

At the very top, the ball is at its maximum height only for an instant. Set the height equal to 20 m: 20t − 5t² = 20 gives t² − 4t + 4 = 0, so t = 2 twice. That is the repeated root, the top of the parabola. Setting the height to 25 m gives no real time at all, because the ball never gets that high. That is the D < 0 case.

When One of the Two Roots Is Rejected

Quadratic equations in word problems often produce two roots, only one of which fits the situation.

Example. The length of a rectangle is 3 m more than its breadth, and its area is 40 m². Find the dimensions.

Let the breadth be x. Then x(x + 3) = 40, so x² + 3x − 40 = 0, which factorises as (x + 8)(x − 5) = 0. So x = 5 or x = −8. A length cannot be negative, so we reject −8. The breadth is 5 m and the length is 8 m.

The mathematics gives two roots, and the problem decides which is meaningful. Always say why you reject one.

Sum and Product of the Roots

Because the roots are α and β, comparing a(x − α)(x − β) with ax² + bx + c gives two useful facts:

• Sum of the roots: α + β = −b/a
• Product of the roots: αβ = c/a

For x² − 5x + 6 = 0, the sum is 5 and the product is 6, which matches the roots 2 and 3. You can use this to check your answer, or to build an equation from given roots.

Finding an Unknown Using the Discriminant

Example. Find the values of k for which 2x² + kx + 3 = 0 has two equal roots.

For equal roots, D = 0. Here a = 2, b = k, c = 3, so D = k² − 4(2)(3) = k² − 24. Set k² − 24 = 0, which gives k = ±√24 = ±2√6.

Notice that this question also has two answers, which is the same pattern again, because k² = 24 is itself a quadratic condition.

Common Slips

• Saying “D = 0 means one root” without adding that it is a repeated root.
• Forgetting that a cannot be zero. If a = 0, the equation is linear.
• Leaving both roots in a word problem without checking that each makes sense.
• Making a sign error in b² − 4ac, especially when b or c is negative. Put brackets around negative numbers.
• Forgetting to bring everything to one side so that the equation equals zero before reading a, b and c.

For the errors that creep into the formula itself, read the common mistakes in the quadratic formula, and for more worked problems, use these important questions with solutions.

Quadratic Equations in Your Syllabus

Quadratic Equations is a chapter of the Class 10 Maths book in both the CBSE and the Karnataka SSLC textbooks. For a general overview, see this blog on exploring quadratic equations. The ideas return in Class 11 with complex roots when D < 0, and they matter for JEE Main and KCET. Confirm the current portions with your own board.

Frequently Asked Questions

Q1. Why does a quadratic equation have two roots?

A quadratic can be written as a product of two linear factors, and setting each factor to zero gives one root. Graphically, a parabola can cut the x-axis at most twice.

Q2. What does the discriminant tell you?

The discriminant D = b² − 4ac tells you the nature of the roots: two distinct real roots if D > 0, two equal real roots if D = 0, and no real roots if D < 0.

Q3. Can a quadratic equation have only one root?

If D = 0, the two roots are equal, so there is one distinct value, which is counted twice as a repeated root.

Q4. Why do we sometimes reject one root in a word problem?

The equation gives two mathematical roots, but a quantity such as a length or a time cannot be negative, so the root that does not fit the situation is rejected.

Q5. What are the sum and product of the roots?

For ax² + bx + c = 0, the sum of the roots is −b/a and the product is c/a.

Two Roots, One Idea

Two factors, two crossings, two signs in the formula: the same fact seen from three directions. Once you see the pattern, you will not need to memorise that a quadratic has two roots. You will be able to explain it.