What is a System of Quadratic Equations?

S6-SA1-0236

Grade Level:

Class 10

AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine

Definition

What is it?

A system of quadratic equations is a set of two or more quadratic equations that share the same variables. The goal is to find the values of these variables that satisfy all equations in the system simultaneously. Think of it like solving multiple puzzles at once, where each puzzle gives you a clue about the same hidden numbers.

Simple Example

Quick Example

Imagine you and your friend are buying samosas and chai. You say, 'Two samosas and one chai cost 50 rupees.' Your friend says, 'One samosa and two chai cost 40 rupees.' If the price of a samosa (x) and chai (y) are related in a quadratic way, you'd have a system of quadratic equations to find their exact prices. For example, if the cost was based on quantities squared, it could be x^2 + y = 50 and x + y^2 = 40.

Worked Example

Step-by-Step

Let's solve a simple system:
Equation 1: y = x^2 + 1
Equation 2: y = 2x + 1

Step 1: Since both equations are equal to 'y', we can set them equal to each other.
x^2 + 1 = 2x + 1

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Step 2: Move all terms to one side to form a standard quadratic equation.
x^2 + 1 - 2x - 1 = 0
x^2 - 2x = 0

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Step 3: Factor out the common term 'x'.
x(x - 2) = 0

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Step 4: Set each factor equal to zero to find the possible values for 'x'.
x = 0 OR x - 2 = 0
x = 0 OR x = 2

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Step 5: Substitute these 'x' values back into either original equation to find the corresponding 'y' values. Let's use y = 2x + 1.
For x = 0: y = 2(0) + 1 = 1
For x = 2: y = 2(2) + 1 = 4 + 1 = 5

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Step 6: The solutions are pairs (x, y).
Solution 1: (0, 1)
Solution 2: (2, 5)

Answer: The solutions to the system are (0, 1) and (2, 5).

Why It Matters

Understanding systems of quadratic equations is crucial for designing everything from satellite trajectories in Space Technology to predicting chemical reactions in Chemistry. Engineers use them to model how bridges bend under weight, and physicists use them to describe projectile motion. These concepts are foundational for careers in AI/ML, engineering, and even medicine, where modeling complex biological systems is key.

Common Mistakes

MISTAKE: Only finding one variable (e.g., just 'x') and forgetting to find the corresponding 'y' values. | CORRECTION: Always substitute the found 'x' values back into one of the original equations to find the complete (x, y) solution pairs.

MISTAKE: Incorrectly combining or subtracting equations, especially when dealing with squared terms or different variable powers. | CORRECTION: Ensure you are combining like terms correctly. If using substitution, make sure to substitute the entire expression accurately.

MISTAKE: Forgetting that a quadratic equation can have two distinct solutions for a variable, leading to potentially two different solution pairs for the system. | CORRECTION: After solving for one variable (e.g., 'x') from the resulting quadratic, always consider both possible roots (solutions) and find the corresponding 'y' values for each.

Practice Questions

Try It Yourself

QUESTION: Solve the system: Equation 1: y = x^2 - 3 | Equation 2: y = x - 1 | ANSWER: (2, 1) and (-1, -2)

QUESTION: Find the solutions for the system: Equation 1: x^2 + y = 10 | Equation 2: y = 6 | ANSWER: (2, 6) and (-2, 6)

QUESTION: A farmer wants to fence a rectangular plot. The area of the plot is given by y = x^2 - 4x + 7, where x is the length. If the perimeter of the plot is 2x + 2y = 18, find the possible values for x and y. | ANSWER: (x, y) = (3, 4)

MCQ

Quick Quiz

Which of the following is NOT a system of quadratic equations?

y = x^2 + 5 and y = 2x - 1

x^2 + y^2 = 25 and x - y = 7

y = 3x + 2 and y = 5x - 4

x^2 + y = 10 and x + y^2 = 12

The Correct Answer Is:

Option C contains two linear equations, not quadratic ones, as neither x nor y is raised to the power of 2. Options A, B, and D all contain at least one squared variable term.

Real World Connection

In the Real World

Imagine a drone delivering food packets in a crowded Indian city. Its flight path might be modeled by a quadratic equation. If there's a no-fly zone boundary, also described by another quadratic equation, solving a system of these equations helps determine if the drone's path intersects the restricted area. This is vital for safe and efficient drone delivery services like those planned by companies for medical supplies or food.

Key Vocabulary

Key Terms

QUADRATIC EQUATION: An equation where the highest power of the variable is 2, like ax^2 + bx + c = 0 | SYSTEM OF EQUATIONS: A set of two or more equations with the same variables | SOLUTION: The values of the variables that satisfy all equations in the system simultaneously | SUBSTITUTION METHOD: Solving one equation for a variable and plugging that expression into another equation | ELIMINATION METHOD: Adding or subtracting equations to remove one variable

What's Next

What to Learn Next

Great job understanding systems of quadratic equations! Next, you can explore 'Solving Systems of Quadratic Equations Graphically' to visualize these solutions. This will help you understand how curves intersect and reinforce your algebraic problem-solving skills, preparing you for more complex math in Class 11.