Solutions by Quadratic Formula MCQs Quiz | Class 10

Understanding Solutions by Quadratic Formula

A quadratic equation is a polynomial equation of the second degree, meaning it contains at least one term in which the unknown variable is raised to the second power. The standard form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are real numbers and a is not equal to 0.

The quadratic formula is a powerful tool used to find the values of the variable (often ‘x’) that satisfy any quadratic equation. It provides a direct method to find the roots, or solutions, of the equation, even when factorization methods are difficult or impossible to apply.

The Quadratic Formula:

The roots of a quadratic equation ax² + bx + c = 0 are given by the formula:

Key Components and Their Significance:

1. Coefficients (a, b, c): These are the numerical values attached to the terms in the standard form of the quadratic equation. ‘a’ is the coefficient of x², ‘b’ is the coefficient of x, and ‘c’ is the constant term. Correctly identifying these values, including their signs, is the first critical step.
2. Discriminant (D): The expression inside the square root, b² – 4ac, is known as the discriminant. Its value determines the nature of the roots of the quadratic equation:
   • If D > 0: The equation has two distinct real roots.
   • If D = 0: The equation has two equal real roots (a repeated root).
   • If D < 0: The equation has no real roots (it has two complex conjugate roots).

Steps to Use the Quadratic Formula (Solving & Simplification):

1. Standard Form: Always ensure the quadratic equation is arranged in the standard form ax² + bx + c = 0. If it’s not, rearrange the terms appropriately.
2. Identify Coefficients: Carefully identify the numerical values of a, b, and c, including their positive or negative signs.
3. Calculate Discriminant: Compute the value of the discriminant, D = b² – 4ac. This step helps in predicting the type of solutions you will get.
4. Substitute and Simplify: Substitute the identified values of a, b, c, and the calculated D into the quadratic formula. Perform all arithmetic operations with precision, particularly when dealing with the square root and division.
5. Find the Roots: Finally, calculate the two possible values for x. One value uses the ‘+’ sign before the square root, and the other uses the ‘-‘ sign.

Example of Solving and Simplification:

Let’s solve the equation 2x² + 5x – 3 = 0 using the quadratic formula.

Here, a = 2, b = 5, c = -3.

1. Calculate Discriminant:
   D = b² – 4ac = (5)² – 4(2)(-3) = 25 – (-24) = 25 + 24 = 49.
   Since D > 0, we expect two distinct real roots.
2. Apply the Formula:
   x = [-b ± √D] / 2a
   x = [-5 ± √49] / 2(2)
   x = [-5 ± 7] / 4
3. Find the Roots:
   For the ‘+’ sign: x = [-5 + 7] / 4 = 2 / 4 = 1/2
   For the ‘-‘ sign: x = [-5 – 7] / 4 = -12 / 4 = -3

Thus, the solutions to the equation 2x² + 5x – 3 = 0 are x = 1/2 and x = -3.

Quick Revision Points:

• The standard form of a quadratic equation is ax² + bx + c = 0.
• The quadratic formula is x = [-b ± √(b² – 4ac)] / 2a.
• The discriminant (D = b² – 4ac) dictates the nature of the roots.
• Always double-check the signs of a, b, and c when identifying them.
• The quadratic formula is a universal method for solving all quadratic equations.

Practice Questions:

• Solve x² – 7x + 12 = 0 using the quadratic formula.
• Find the discriminant of 3x² – 4x + 1 = 0.
• For what value of m does 4x² – 12x + m = 0 have equal roots?
• Determine the nature of the roots for 5x² + 2x + 1 = 0.
• Find the roots of x² + 6x + 9 = 0.