What is the Relationship Between Zeroes and Coefficients?

S3-SA1-0110

Grade Level:

Class 7

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

Definition

What is it?

The relationship between zeroes and coefficients helps us understand how the roots (zeroes) of a polynomial equation are connected to the numbers (coefficients) in front of its variables. For a quadratic equation like ax^2 + bx + c = 0, there are simple formulas linking its zeroes to 'a', 'b', and 'c'.

Simple Example

Quick Example

Imagine you have a quadratic equation that describes how many runs a cricket team needs to score in 'x' overs to win, like x^2 - 5x + 6 = 0. The zeroes of this equation would tell you the specific number of overs where the run rate equation balances out. The coefficients (1, -5, 6) are the numbers in the equation, and they hold the secret to finding those specific overs without even solving the whole equation!

Worked Example

Step-by-Step

Let's find the relationship for the quadratic equation x^2 - 7x + 10 = 0.
---Step 1: Identify the coefficients. Here, a = 1 (coefficient of x^2), b = -7 (coefficient of x), and c = 10 (constant term).
---Step 2: Find the zeroes of the equation. We can factorize: x^2 - 2x - 5x + 10 = 0 => x(x-2) - 5(x-2) = 0 => (x-2)(x-5) = 0. So, the zeroes are x = 2 and x = 5.
---Step 3: Calculate the sum of the zeroes. Sum = 2 + 5 = 7.
---Step 4: Calculate the product of the zeroes. Product = 2 * 5 = 10.
---Step 5: Compare with the formulas. For a quadratic equation ax^2 + bx + c = 0, Sum of zeroes = -b/a. Here, -b/a = -(-7)/1 = 7. This matches our calculated sum.
---Step 6: Product of zeroes = c/a. Here, c/a = 10/1 = 10. This matches our calculated product.
Answer: The sum of zeroes is 7 and the product of zeroes is 10, which matches -b/a and c/a respectively.

Why It Matters

Understanding this relationship is super useful in fields like AI/ML and Data Science to quickly analyze patterns in data. Engineers use it to design structures and predict outcomes, while economists use it to model market trends. It helps scientists and engineers solve complex problems faster and more efficiently.

Common Mistakes

MISTAKE: Forgetting the negative sign in the sum of zeroes formula (-b/a). | CORRECTION: Always remember that the sum of zeroes is -b/a, not just b/a. Pay close attention to the sign of 'b'.

MISTAKE: Confusing 'a', 'b', and 'c' with the zeroes themselves. | CORRECTION: 'a', 'b', and 'c' are the coefficients (numbers in the equation), while the zeroes are the values of 'x' that make the equation true.

MISTAKE: Not arranging the quadratic equation in standard form (ax^2 + bx + c = 0) before identifying coefficients. | CORRECTION: Always rewrite the equation in the standard form first to correctly identify 'a', 'b', and 'c'.

Practice Questions

Try It Yourself

QUESTION: For the equation x^2 - 9x + 14 = 0, what is the sum of its zeroes? | ANSWER: 9

QUESTION: If the product of the zeroes of a quadratic equation is 8 and the equation is x^2 + 2x + k = 0, what is the value of k? | ANSWER: 8

QUESTION: A quadratic equation has zeroes at x = 3 and x = -5. What are the values of -b/a and c/a for this equation? | ANSWER: -b/a = -2, c/a = -15

MCQ

Quick Quiz

For the quadratic equation 3x^2 + 6x - 9 = 0, what is the sum of its zeroes?

-2

-3

The Correct Answer Is:

For 3x^2 + 6x - 9 = 0, a=3, b=6, c=-9. The sum of zeroes is -b/a = -6/3 = -2. So, option B is correct.

Real World Connection

In the Real World

Imagine an app like 'Swiggy' or 'Zomato' trying to optimize delivery routes. They might use equations where zeroes represent optimal stopping points or times. The coefficients in these equations are like the variables they can control, such as driver speed or traffic density. Understanding this relationship helps them quickly find the best solutions to get your food delivered faster!

Key Vocabulary

Key Terms

ZEROES: The values of the variable (usually 'x') that make a polynomial equation equal to zero, also called roots. | COEFFICIENTS: The numerical values multiplying the variables in a polynomial term. | QUADRATIC EQUATION: A polynomial equation of degree 2, typically written as ax^2 + bx + c = 0. | STANDARD FORM: The usual way to write a polynomial, with terms arranged from highest to lowest power of the variable. | POLYNOMIAL: An expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.

What's Next

What to Learn Next

Great job understanding this! Next, you can explore how this relationship extends to cubic polynomials (equations with x^3). This will open up even more ways to solve and understand complex equations, preparing you for higher-level algebra and problem-solving.