Real Roots of Quadratic Equations - CBSE Class 10 Mathematics Quiz

Real Roots of Quadratic Equations in CBSE Class 10 Mathematics

Quadratic equations are polynomial equations of the second degree. These equations are in the form of ax² + bx + c = 0, where a, b, and c are constants. In CBSE Class 10 Mathematics, we learn about the real roots of quadratic equations and various methods to find them.

Solutions of Quadratic Equations

The solutions of a quadratic equation ax² + bx + c = 0 can be found by following these steps:

1. Find the discriminant, which is given by the formula b² - 4ac.
2. If the discriminant is positive, there are two real roots.
3. If the discriminant is zero, there is one real root.
4. If the discriminant is negative, there are two complex roots.

Factors of Quadratic Equations

A quadratic equation ax² + bx + c = 0 can be written as a product of two factors. The factors of a quadratic equation are of the form (x - k), where k is a constant. To find the factors of a quadratic equation, we need to solve the following two equations:

1. ax² - kx + c = 0
2. ax² - kx - c = 0

By taking the difference of these two equations, we can find the value of k, which is given by the formula (2ac - b²) / (4a²).

Methods to Find Real Roots of Quadratic Equations

There are several methods to find the real roots of a quadratic equation, including:

1. Factoring Method: This method involves finding the factors of the quadratic equation and solving the resulting equations to find the real roots.

2. Square Root Method: This method involves taking the square root of the coefficients of the quadratic equation and solving the resulting equations to find the real roots.

3. Quadratic Formula: This method involves using the quadratic formula, which is given by the formula x = (-b ± √(b² - 4ac)) / (2a).

Examples

Example 1: Find the real roots of the quadratic equation 2x² - 5x + 3 = 0.

Solution:

1. The discriminant is given by (-5)² - 4(2)(3) = 25 - 24 = 1.
2. Since the discriminant is positive, there are two real roots.
3. The factors of the quadratic equation are (2x - 3) and (2x - 1).
4. Solving the equations 2x² - 3x = 3 and 2x² - 1x = 3, we get x = 1 and x = 1.

Example 2: Find the real roots of the quadratic equation 3x² - 5x + 2 = 0.

Solution:

1. The discriminant is given by (-5)² - 4(3)(2) = 25 - 24 = 1.
2. Since the discriminant is positive, there are two real roots.
3. The factors of the quadratic equation are (3x - 2) and (3x - 1).
4. Solving the equations 3x² - 2x = 2 and 3x² - 1x = 2, we get x = 2 and x = 2.

Practice Questions

1. Find the real roots of the quadratic equation x² - 5x + 6 = 0.
2. Find the real roots of the quadratic equation 2x² - 3x + 1 = 0.
3. Find the real roots of the quadratic equation 4x² + 5x + 1 = 0.

Conclusion

In CBSE Class 10 Mathematics, we learn about the real roots of quadratic equations and various methods to find them. The solutions of a quadratic equation can be found by calculating the discriminant, finding the factors of the equation, or using the quadratic formula. The real roots of a quadratic equation can be found by solving the resulting equations or using the quadratic formula.