What is Equal Roots of a Quadratic Equation?

S3-SA1-0290

Grade Level:

Class 6

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

When you solve a quadratic equation, you usually find two answers for 'x'. If both these answers are exactly the same, then the quadratic equation is said to have 'equal roots'. It means the equation has only one unique solution that appears twice.

Simple Example

Quick Example

Imagine you are trying to find two numbers that multiply to 9 and add up to 6. The only number that works for both is 3 (3 x 3 = 9 and 3 + 3 = 6). Here, the 'root' or solution is 3, and it's the only one, so it's an 'equal root'.

Worked Example

Step-by-Step

Let's check the quadratic equation: x^2 - 6x + 9 = 0

Step 1: We can try to factorise this equation. We need two numbers that multiply to +9 and add up to -6.
---Step 2: The numbers are -3 and -3. So, we can write the equation as (x - 3)(x - 3) = 0.
---Step 3: For the product of two terms to be zero, at least one of them must be zero.
---Step 4: So, x - 3 = 0 OR x - 3 = 0.
---Step 5: Solving for x, we get x = 3 from the first part.
---Step 6: And x = 3 from the second part.
---Step 7: Both solutions for x are 3. Since they are the same, the equation has equal roots.
Answer: The roots are 3 and 3, which are equal.

Why It Matters

Understanding equal roots helps engineers design stable structures and computer scientists optimize algorithms. Data scientists use these concepts to model trends and predict outcomes more accurately. It's a foundational idea for solving complex problems in AI and physics.

Common Mistakes

MISTAKE: Thinking 'equal roots' means no solution or only one solution. | CORRECTION: Equal roots means there are two solutions, but both are identical. It's one unique value repeated.

MISTAKE: Forgetting that the discriminant (b^2 - 4ac) must be exactly zero for equal roots. | CORRECTION: Always check the discriminant. If b^2 - 4ac = 0, then the roots are equal.

MISTAKE: Assuming equal roots only happen with perfect square trinomials. | CORRECTION: While perfect square trinomials always lead to equal roots, it's the discriminant being zero that is the true condition, which can be achieved in other forms too.

Practice Questions

Try It Yourself

QUESTION: Does the equation x^2 - 4x + 4 = 0 have equal roots? | ANSWER: Yes

QUESTION: Find the value of 'k' for which the quadratic equation x^2 + 10x + k = 0 has equal roots. | ANSWER: k = 25

QUESTION: A farmer wants to fence a rectangular plot. The length is 'x' meters and the width is 'x-2' meters. If the area is 1 square meter, does the equation representing this situation have equal roots? (Hint: Form the equation and check its roots) | ANSWER: No, the equation is x^2 - 2x - 1 = 0, which does not have equal roots.

MCQ

Quick Quiz

Which of the following conditions ensures a quadratic equation ax^2 + bx + c = 0 has equal roots?

b^2 - 4ac > 0

b^2 - 4ac < 0

b^2 - 4ac = 0

b^2 - 4ac is a perfect square

The Correct Answer Is:

For a quadratic equation to have equal roots, its discriminant (b^2 - 4ac) must be exactly zero. This means there is only one distinct real solution.

Real World Connection

In the Real World

When designing a bridge or a building, engineers use quadratic equations to calculate stress and load distribution. If a critical calculation results in an equation with equal roots, it might indicate a unique optimal design point or a specific condition where the structure is perfectly balanced, like finding the exact point for a support pillar to prevent sway.

Key Vocabulary

Key Terms

QUADRATIC EQUATION: An equation with the highest power of the variable as 2, like ax^2 + bx + c = 0. | ROOTS: The solutions or values of the variable that make the equation true. | DISCRIMINANT: The part of the quadratic formula, b^2 - 4ac, which tells us about the nature of the roots. | FACTORISATION: Breaking down an expression into a product of simpler terms.

What's Next

What to Learn Next

Great job learning about equal roots! Next, you should explore 'Real and Distinct Roots' and 'No Real Roots' of a quadratic equation. This will complete your understanding of all types of solutions a quadratic equation can have and how the discriminant tells you everything.