How to Solve Quadratic Equation Questions Step by Step?
The concept of quadratic equation questions is a central part of secondary mathematics, featuring widely in school exams, Olympiads, and competitive entrances. Understanding how to solve quadratic equation questions quickly and accurately will boost both your confidence and scores, whether you’re preparing for board exams or tests like JEE, NDA, or NEET. This page will guide you through definitions, formulas, stepwise solving methods, real-life applications, and speed tricks, all presented in a simple, mobile-friendly format—perfect for self-study or revision on-the-go.
What Is a Quadratic Equation Question?
A quadratic equation question typically asks you to solve an equation of the form ax² + bx + c = 0, where a ≠ 0, and a, b, c are constants. Quadratic equations pop up in various contexts, including algebraic manipulation, word problems, and practical applications in physics, engineering, and everyday scenarios. You'll practice identifying roots, factoring polynomials, and using different solving techniques.
Key Formula for Quadratic Equation Questions
To solve any quadratic equation, the most reliable formula is:
\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
Here, “b²-4ac” is known as the discriminant. This determines the nature of the roots of the equation (real, equal, or complex).
Quadratic Equations in Real Life & Entrance Exams
Quadratic equation questions are not just for maths class. You'll encounter them in physics (projectile motion, area), business calculations (profit and loss), engineering, and computer science. In exams like JEE Main, NDA, and banking tests, they are a regular feature. Vedantu’s interactive sessions help students master these questions using clear methods and practical speed strategies.
Three Main Methods to Solve Quadratic Equation Questions
| Method | How It Works | When to Use |
|---|---|---|
| Factoring | Express as (x + p)(x + q) = 0, then solve for x. | Simple equations, integers roots |
| Completing the Square | Rewrite ax² + bx + c = 0 as a perfect square trinomial. | Good for learning, special forms |
| Quadratic Formula | Plug values into \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \) | For any quadratic equation |
Step-by-Step Solution Example
Sample Question: Solve the quadratic equation \( x^2 - 5x + 6 = 0 \)
1. Write it as \( x^2 - 5x + 6 = 0 \ )2. Factor: Find two numbers that multiply to 6 and add to -5. They are -2 and -3.
3. Write: \( (x - 2)(x - 3) = 0 \ )
4. Set each factor to zero:
5. Final Answer: The roots are x = 2 and x = 3.
Solved Quadratic Equation Questions with Detailed Steps
Practice is key to mastering quadratic equations. Here are two more solved examples using different methods:
Q1: Solve \( 2x^2 - 4x + 2 = 0 \) using the quadratic formula.
1. Identify a = 2, b = -4, c = 22. Find the discriminant: D = (-4)² - 4·2·2 = 16 - 16 = 0
3. Substitute into the formula:
4. Root: Both roots are the same: x = 1.
Q2: Solve \( x^2 + 4x + 3 = 0 \) by factoring.
1. Factors of 3 that add up to 4: 1 and 32. Write as \( (x + 1)(x + 3) = 0 \)
3. Roots: x = -1, x = -3
Word Problems on Quadratic Equations
Many real-life situations can be converted into quadratic equation questions. For example,
Q: The product of two consecutive numbers is 56. What are the numbers?
1. Let the numbers be x and x+1.2. Set up the equation: x(x+1) = 56 → \( x^2 + x - 56 = 0 \)
3. Factor: (x + 8)(x - 7) = 0
4. Roots: x = -8 or x = 7
5. Consecutive numbers: (-8, -7) or (7, 8)
Speed Trick or Vedic Shortcut
Here is a shortcut when using the quadratic formula: If the coefficient 'a' is 1 (i.e., x² + bx + c = 0), directly factor the number 'c' whose factors add to 'b'. This can save time, especially in MCQ-based exams.
More advanced tips are taught step-wise in Quadratics topic revision classes at Vedantu.
Frequent Errors and Misunderstandings
- Forgetting that 'a' must not be zero in ax² + bx + c = 0 (otherwise it’s not quadratic).
- Mixing up the plus/minus sign in the quadratic formula.
- Ignoring negative roots in word problems (when negative answers don’t make sense in context).
- Wrongly identifying the discriminant.
- Calculation slip-ups when squaring or multiplying terms.
Relation to Other Concepts
Understanding quadratic equation questions helps with polynomial equations, algebraic identities, and graphing parabolas. Knowing the difference between linear and quadratic equations makes topic selection in MCQs easier.
Try These Yourself
- Solve: \( x^2 + 6x + 9 = 0 \)
- Solve by quadratic formula: \( 3x^2 - 12x + 12 = 0 \)
- Find the discriminant and type of roots for \( x^2 - 2x + 3 = 0 \)
- Frame a word problem based on the formula \( ax^2 + bx + c = 0 \).
Classroom Tip
A quick memory aid: “Plus or Minus, b² minus 4ac, over 2a.” If you ever forget the quadratic formula, just recall this rhyme used by Vedantu’s teachers in live doubt sessions.
Wrapping It All Up
We learned how to solve quadratic equation questions using stepwise illustrations, learned common errors, and practiced both mathematical and real-world problems. Consistent practice, as given on Quadratic Equations for Class 10, will make these questions second nature. Keep revising, and you’ll be ready to solve any quadratic equation challenge on your next test!
