How to Solve Quadratic Equations Step by Step for Class 10
The concept of Quadratic Equation for Class 10 is essential in mathematics and helps students solve real-life problems as well as exam-based questions efficiently. Mastering this topic is key for success in CBSE and other boards, as well as for higher studies in algebra and beyond.
Understanding Quadratic Equation for Class 10
A Quadratic Equation for Class 10 is a type of polynomial equation in one variable where the highest power of the variable is 2. It is usually written in the standard form ax² + bx + c = 0, where a, b, and c are real numbers and a ≠ 0. This core concept appears in algebra, board exam questions, and real-world areas like area calculation and speed problems.
Formula Used in Quadratic Equation for Class 10
The key formulae for solving a quadratic equation for class 10 are:
Standard form: ax² + bx + c = 0
Quadratic Formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
Here, “a” is the coefficient of x², “b” is the coefficient of x, and “c” is the constant term.
Here’s a helpful table to understand Quadratic Equation for Class 10 more clearly:
Key Points Table for Quadratic Equation for Class 10
| Term | Meaning/Value | Significance |
|---|---|---|
| Quadratic Equation | ax² + bx + c = 0 | Standard Form |
| Quadratic Formula | \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) | To find roots |
| Discriminant (D) | b² - 4ac | Nature of Roots |
| Sum of Roots | -b/a | Exam Questions |
| Product of Roots | c/a | Exam Questions |
This table helps you quickly recall all the important points in Quadratic Equations for Class 10.
Step-by-Step Solution: How to Solve a Quadratic Equation
You can solve a Quadratic Equation for Class 10 using the following methods:
a. Rewrite the equation ax² + bx + c = 0 in standard form.
b. Find two numbers such that their sum is b and product is ac.
c. Split the middle term and factor.
d. Set each factor equal to zero and solve for x.
a. Write the equation in standard form ax² + bx + c = 0.
b. Identify values of a, b, c.
c. Use the formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
d. Calculate the discriminant D = b² - 4ac.
e. Find the value(s) of x.
a. Take all terms to one side.
b. Divide by a (if a ≠ 1).
c. Add and subtract (b/2a)².
d. Rewrite as perfect square and take square roots.
e. Solve for x.
Worked Example – Solving a Quadratic Equation for Class 10
Example: Solve the quadratic equation \(2x^2 - 7x + 3 = 0\) using the quadratic formula.
2. Identify \(a = 2\), \(b = -7\), \(c = 3\)
3. Calculate the discriminant: \(D = (-7)^2 - 4 \times 2 \times 3 = 49 - 24 = 25\)
4. Substitute in the quadratic formula:
\( x = \frac{-(-7) \pm \sqrt{25}}{2 \times 2} \) \( x = \frac{7 \pm 5}{4} \)
5. Calculate the roots:
\( x_1 = \frac{7 + 5}{4} = \frac{12}{4} = 3 \)
\( x_2 = \frac{7 - 5}{4} = \frac{2}{4} = 0.5 \)
Final Answer: The roots are x = 3 and x = 0.5.
Practice Problems
- Solve: \( x^2 + 5x + 6 = 0 \)
- Find the roots of \( 3x^2 + 4x - 7 = 0 \)
- Solve by factorisation: \( x^2 - 3x - 10 = 0 \)
- What is the nature of roots for \( x^2 + 4x + 5 = 0 \)?
- Form a quadratic equation whose roots are 2 and -3.
Common Mistakes to Avoid
- Not converting to standard form before applying methods.
- Calculation errors in the discriminant (b² - 4ac).
- Forgetting both + and – while using “± ” in Quadratic Formula.
- Mixing up linear and quadratic equations. Review linear equations here.
Real-World Applications
Quadratic equations for Class 10 are useful in real-world problems like finding areas of plots, predicting the path of an object thrown up, and profit calculations in businesses. Vedantu strives to help make these concepts practical and accessible for students, ensuring smooth learning.
Revision Tips & NCERT Insights
- Always write the equation in standard form before solving.
- Check the value of discriminant to decide the nature of roots. Refer to more details on Discriminant.
- Practice word problems and extra questions from NCERT and sample papers.
- Download revision notes or formula sheets from Vedantu's Maths Formulas for Class 10 page.
Helpful Internal Links
- Quadratic Equations (Concepts & Properties)
- Polynomial (Understanding Polynomial Equations)
- Solution of Quadratic Equation in Complex Number System
- Complex Numbers and Quadratic Equations
- Quadratic Equation Questions (Practice Problems)
- Maths Formulas for Class 10 (All Topics Quick Revision)
- Completing the Square
- Linear Equations in One Variable
- Algebraic Equations
- Quadratic Equations (General Topic Page)
- Maths Equations (General Overview)
We explored the idea of Quadratic Equation for Class 10, the methods to solve it, worked examples, important formulas, and application tips for exams. Keep practicing on Vedantu to gain more confidence and clarity on quadratic equations and boost your board exam performance.
FAQs on Quadratic Equation for Class 10 – Definition, Formula & Practice
1. What is a quadratic equation for Class 10?
A quadratic equation for Class 10 is a polynomial equation of degree two, usually expressed as ax² + bx + c = 0, where a, b, and c are real numbers with a ≠ 0. This equation forms the basis of many problems in algebra and helps in understanding parabolas, roots, and their applications.
2. What is the quadratic formula for class 10?
The quadratic formula is a universal method to find the roots of any quadratic equation of the form ax² + bx + c = 0. It is given by:
x = [-b ± √(b² - 4ac)] / 2a
Substituting values of a, b, and c in the formula yields the roots, which may be real or complex depending on the discriminant.
3. How do you solve quadratic equations for board exams?
To solve quadratic equations effectively for Class 10 board exams, follow these common methods:
1. Factorisation Method: Express the quadratic polynomial as a product of two binomials and find roots by setting each factor to zero.
2. Completing the Square: Rewrite the quadratic into a perfect square trinomial and solve for the variable.
3. Quadratic Formula: Use the formula x = [-b ± √(b² - 4ac)] / 2a to find roots directly when factoring is difficult.
Practicing these steps with examples improves accuracy and speed for exams.
4. Can you list 10 examples of quadratic equations for Class 10?
Here are 10 typical quadratic equations that align with the Class 10 syllabus:
1. x² - 5x + 6 = 0
2. 2x² + 3x - 5 = 0
3. x² + 4x + 4 = 0
4. 3x² - 2x + 1 = 0
5. 4x² - 7x + 3 = 0
6. x² - 9 = 0
7. 5x² + 6x = 0
8. x² + x - 12 = 0
9. 2x² - 3x - 2 = 0
10. x² + 2x + 1 = 0.
Practicing these examples covers a broad range of factorable and non-factorable quadratics typical in exams.
5. What are the quadratic equation formulas for Class 10?
The essential quadratic equation formulas for Class 10 include:
• The Standard Form: ax² + bx + c = 0
• Quadratic Formula: x = [-b ± √(b² - 4ac)] / 2a
• Discriminant: D = b² - 4ac, which determines the nature of roots
• Sum of Roots: α + β = -b/a
• Product of Roots: αβ = c/a
These formulas are fundamental tools to solve, analyze, and understand the properties of quadratic equations.
6. How to download a quadratic equation worksheet for Class 10?
To download a quadratic equation worksheet for Class 10, visit Vedantu’s Maths topic pages or official resource sections providing free PDF downloads. These worksheets include:
• Multiple choice questions
• Solved examples
• Extra practice problems with solutions
Simply click the worksheet download links on the topic page to save the PDF for offline mobile or desktop revision.
7. Why is the quadratic formula sometimes confusing for students?
The quadratic formula can be confusing due to:
• Complex symbols and fractions leading to calculation errors.
• Misunderstanding the discriminant and its impact on roots.
• Forgetting to correctly substitute and simplify the values of a, b, and c.
To overcome this, practice careful stepwise substitution and familiarize yourself with the formula’s structure.
8. Why do students confuse quadratic equations with linear ones?
Students often confuse quadratic and linear equations because:
• Both involve variables and algebraic expressions.
• Lack of clear understanding that quadratic equations have the variable raised to the power 2 (degree 2), while linear equations have degree 1.
• Similar equation formats in early examples cause misunderstanding.
Emphasizing the highest degree term and graphical differences (parabola vs. line) helps clarify this distinction.
9. How do quadratic equations relate to real-life Class 10 applications?
Quadratic equations relate to real-life applications in Class 10 through problems involving:
• Calculating areas of geometric shapes like rectangles and triangles.
• Determining the speed and trajectory in physics problems.
• Analyzing revenue, profit, and loss in business scenarios.
• Modelling natural phenomena that follow parabolic curves.
Understanding and solving quadratic equations empowers students to connect math theory with practical problems.
10. Why is it important to practice extra questions for board exams?
Practicing extra quadratic equation questions before board exams is important because:
• It builds confidence in applying multiple solution methods.
• Helps familiarize with various question patterns including tricky and lengthy problems.
• Enhances problem-solving speed and accuracy under exam conditions.
• Identifies and strengthens weak concepts.
Consistent practice ensures readiness and helps achieve better exam scores.
11. What mistakes are common in quadratic equation word problems?
Common mistakes in quadratic equation word problems include:
• Incorrect translation of the problem statement into the quadratic form.
• Ignoring units or mixing different units.
• Missing negative signs or errors in the coefficients.
• Not checking the discriminant before solving.
• Forgetting to verify roots in the context of the problem.
Careful reading, stepwise equation formation, and rechecking answers reduce these errors.
