What Is a Quadratic Equation Standard Form Formula and How to Solve with Examples
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 Maths Complete Guide
1. What is a quadratic equation in Class 10?
A quadratic equation is a polynomial equation of degree 2 in one variable, written in the form ax² + bx + c = 0 where a ≠ 0.
- Here, a, b, c are real numbers.
- x is the variable.
- The highest power of x is 2, which makes it quadratic.
2. What is the standard form of a quadratic equation?
The standard form of a quadratic equation is ax² + bx + c = 0, where a ≠ 0.
- a is the coefficient of x².
- b is the coefficient of x.
- c is the constant term.
3. What is the quadratic formula for Class 10?
The quadratic formula is x = (−b ± √(b² − 4ac)) / 2a.
- It is used to find the roots of ax² + bx + c = 0.
- The term b² − 4ac is called the discriminant.
4. What is the discriminant in a quadratic equation?
The discriminant of a quadratic equation is D = b² − 4ac.
- If D > 0, the roots are real and distinct.
- If D = 0, the roots are real and equal.
- If D < 0, the roots are complex (no real roots).
5. How do you solve a quadratic equation by factorization?
To solve a quadratic equation by factorization, split the middle term and factor the expression into two linear factors.
- Step 1: Write the equation in ax² + bx + c = 0 form.
- Step 2: Find two numbers whose product is ac and sum is b.
- Step 3: Split the middle term and factor by grouping.
- Step 4: Set each factor equal to zero and solve.
6. How do you find the roots of a quadratic equation?
The roots of a quadratic equation are the values of x that satisfy ax² + bx + c = 0.
- Use factorization if the equation is easily factorable.
- Use the quadratic formula for all types of equations.
- You can also use the completing the square method.
7. What are real and equal roots in a quadratic equation?
A quadratic equation has real and equal roots when the discriminant D = 0.
- This means b² − 4ac = 0.
- Both roots are the same value.
8. What is the relationship between the roots and coefficients of a quadratic equation?
The relationship between roots and coefficients of ax² + bx + c = 0 is given by α + β = −b/a and αβ = c/a.
- α and β are the roots.
- The sum of roots depends on coefficient b.
- The product of roots depends on constant term c.
9. How do you form a quadratic equation when the roots are given?
If the roots are α and β, the quadratic equation is x² − (α + β)x + αβ = 0.
- Step 1: Find the sum α + β.
- Step 2: Find the product αβ.
- Step 3: Substitute into the formula.
10. What are some real-life applications of quadratic equations?
Quadratic equations are used to model situations involving area, speed, profit, and projectile motion.
- Finding dimensions of a rectangle when area is given.
- Calculating maximum profit in business problems.
- Determining the path of a ball thrown in the air.
