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What is the Sum of Roots of a Quadratic Equation?
Grade Level:
Class 6
AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering
Definition
What is it?
The 'sum of roots' of a quadratic equation is simply the total you get when you add up the two solutions (or 'roots') of that equation. A quadratic equation is a special type of equation where the highest power of the variable is 2, like x^2. These roots are the values of the variable that make the equation true.
Simple Example
Quick Example
Imagine you have two friends, Rahul and Priya, who each scored points in a game. If Rahul scored 5 points and Priya scored 3 points, their total score (sum) is 5 + 3 = 8 points. Similarly, if the 'roots' of an equation are 5 and 3, their sum is 8.
Worked Example
Step-by-Step
Let's find the sum of roots for the quadratic equation: x^2 - 7x + 10 = 0.
Step 1: Identify the coefficients. A standard quadratic equation is ax^2 + bx + c = 0. Here, a = 1 (coefficient of x^2), b = -7 (coefficient of x), and c = 10 (the constant term).
---Step 2: Use the formula for the sum of roots. The sum of roots is given by the formula -b/a.
---Step 3: Substitute the values of 'b' and 'a' into the formula. Sum of roots = -(-7) / 1.
---Step 4: Simplify the expression. -(-7) becomes +7.
---Step 5: Calculate the final value. Sum of roots = 7 / 1 = 7.
So, the sum of the roots for the equation x^2 - 7x + 10 = 0 is 7.
Why It Matters
Understanding the sum of roots helps engineers design structures, economists predict market trends, and computer scientists develop algorithms for AI. It's a foundational concept used in many fields, from building bridges to making your favorite apps smarter, and even in careers like data analysis and cryptography.
Common Mistakes
MISTAKE: Forgetting the negative sign in the -b/a formula. Students often use b/a instead of -b/a. | CORRECTION: Always remember the formula is -b/a. If 'b' itself is negative, like -7, then -b becomes -(-7) which is +7.
MISTAKE: Confusing 'b' and 'c' or 'a' in the formula. Students might pick the wrong coefficient. | CORRECTION: Remember 'a' is with x^2, 'b' is with x, and 'c' is the constant term. Always match them correctly.
MISTAKE: Not simplifying the equation to standard form (ax^2 + bx + c = 0) first. For example, if it's x^2 + 10 = 7x. | CORRECTION: Always rearrange the equation so all terms are on one side and it equals zero before identifying a, b, and c.
Practice Questions
Try It Yourself
QUESTION: Find the sum of roots for the equation x^2 - 5x + 6 = 0. | ANSWER: 5
QUESTION: What is the sum of roots for the equation 2x^2 + 8x - 10 = 0? | ANSWER: -4
QUESTION: If the equation is 3x^2 - 12 = 0, what is the sum of its roots? (Hint: What is the value of 'b' here?) | ANSWER: 0
MCQ
Quick Quiz
For the quadratic equation 4x^2 - 12x + 9 = 0, what is the sum of its roots?
3
-3
12
9
The Correct Answer Is:
A
The formula for the sum of roots is -b/a. Here, a = 4 and b = -12. So, -b/a = -(-12)/4 = 12/4 = 3.
Real World Connection
In the Real World
Imagine a cricket analyst at IPL using equations to predict a batsman's performance based on different game conditions. The roots of these equations could represent specific outcomes, and their sum might help calculate overall risk or success probabilities. This is similar to how data scientists at companies like Swiggy use math to optimize delivery routes or predict demand.
Key Vocabulary
Key Terms
QUADRATIC EQUATION: An equation where the highest power of the variable is 2, like ax^2 + bx + c = 0. | ROOTS: The solutions or values of the variable that make the equation true. | COEFFICIENT: A number multiplied by a variable in an algebraic expression. | CONSTANT TERM: A term in an algebraic expression that does not contain any variables.
What's Next
What to Learn Next
Great job understanding the sum of roots! Next, you should explore 'What is the Product of Roots of a Quadratic Equation?' It uses a similar approach and is another key property that helps us understand quadratic equations better. Keep learning!
