[Maths Class Notes] on Quadratic Equation Questions Pdf for Exam

Quadratic equations are an important part of algebra, and as students, we must all be familiar with their definition and the ways of solving quadratic equation problems. In this article, we are going to familiarize the students with all the concepts surrounding quadratic equations and the methods of solving problems related to this topic. A short definition of a quadratic equation would be: a quadratic equation is a second-degree polynomial, which we represent as ‘ax²+ bx + c’ in general.

In this representation, a cannot be equal to 0 and b,c are known as coefficients and are constant by nature. With this basic introduction, let’s move forward with a formal definition, formulae and detailed solutions to quadratic equation questions to enable better understanding.

Definition of Quadratic Equations

A quadratic equation is a polynomial where the highest power of the variable is 2. We generally represent it as ax² + bx + c. Here a, b and c are real numbers or constants, and x is the variable. In this case, the value of a cannot be 0 as that would remove the x² term, and the equation won’t be quadratic after that.

A quadratic equation is an equation of second degree with more than two terms. It means that at least one of the terms of the equation is squared. In the above-given equation. The answer to the equation also known as the roots of the equation is the value of the “x”. The value of the “x” has to satisfy the equation. 

Some examples of quadratic equations can be as follows:

• 56x² + ⅔ x + 1, where a = 56, b = ⅔ and c = 1.

• -4/3 x² + 64x – 30, where a = -4/3, b = 64 and c = -30.

Roots of a Quadratic Equation

To solve basic quadratic equation questions or any quadratic equation problems, we need to solve the equation. Solving quadratic equations gives us the roots of the polynomial. The roots of the equation are the values of x at which ax² + bx + c = 0. Since a quadratic equation is a polynomial of degree 2, we obtain two roots in this case.

There are several methods for solving quadratic equation problems, as we can see below:

Quadratic Equation Formula

So what is the quadratic equation formula? The quadratic equation formula or the Sridharacharya Formula is a method for finding out the roots of two-degree polynomials. This formula helps solve quadratic equation problems. The formula is as given below:

[ x = frac{-b pmsqrt{b^{2}-4ac}}{2a} ]

Where x represents the roots of the equation and (b²−4ac) is the discriminant.

By finding out the value of the discriminant, we can predict the nature of the roots. There are three possibilities with three different implications:

• Two distinct roots which are real, if b² – 4ac > 0.

• Two real roots equal in magnitude, if b² – 4ac = 0.

• Imaginary roots or absence of real roots if b² – 4ac < 0.

Now that the basic principles of quadratic equations are clear, we will move on to some solved examples. But before that, let us list some quadratic equation questions for the students to solve.

Quadratic Equation Practice Questions

The following are a list of questions for you to solve once you have gone through the quadratic equation questions and answers in the solved examples section:

1. Find the determinant of the following quadratic equations: 2x² + 3x + 6, 70x² + 49 + 14, ⅔ y² + 63y + 42.

2. Find the roots of the following quadratic equations: x² – 45x + 324, 2x² – 22x + 42, ½ x² + 2x + 4.

3. The product of two consecutive numbers is 420, and their sum is 41. Find the numbers.

Before solving these, let’s check out the solved examples with questions and answers on the quadratic equation.

Solved Examples

Q1. Solve x² + 5x + 6 = 20 by factorization method.

Answer: The given polynomial or quadratic equation is

x² + 5x + 6 = 20

Solving by factorization method, 

x² + 5x + 6 – 20 = 0.

or, x² + 5x – 14 = 0

or, x² – 2x + 7x – 14 = 0

or, x(x – 2) +7(x – 2) = 0

or, (x – 2)(x + 7) = 0

or, (x – 2) = 0, (x + 7) = 0 

or, x = +2, -7.

Q2. Solve 2x² – 5x + 3 using the quadratic equation formula.

Answer: The quadratic equation formula is:

[ x = frac{-bpmsqrt{b^{2}-4ac}}{2a} ]

The determinant or b²-4ac = (-5)² – 4 × 3 × 2 = 25 – 24 = 1

[ sqrt{b^{2} – 4ac} = 1 ]

Therefore, [ x= frac{-(-5)pm1}{2times2} ]

[ x = frac{5+1}{4} = frac{6}{4} = frac{3}{2} ]

[ x = frac{5-1}{4} = frac{4}{4} = 1 ]

Thus the roots of the equation are 3/2 and 1.

This is all about the roots of quadratic equations and their formulas. Learn the formulas and find out how they are used to derive the roots of an equation easily.