Quadratic Equation Questions

Quadratic equation questions are provided here for Class 10 students. A quadratic equation is a second-degree polynomial which is represented as ax² + bx + c = 0, where a is not equal to 0. Here, a, b and c are constants, also called coefficients and x is an unknown variable. Also, learn Quadratic Formula here.

Solving the problems based on quadratics will help students to understand the concept very well and also to score good marks in this section. All the questions are solved here step by step with a detailed explanation. In this article, we will give the definition and important formula for solving problems based on quadratic equations. The questions given here is in reference to the CBSE syllabus and NCERT curriculum. 

Definition of Quadratic Equation

Usually, the quadratic equation is represented in the form of ax²+bx+c=0, where x is the variable and a,b,c are the real numbers & a ≠ 0. Here, a and b are the coefficients of x² and x, respectively. So, basically, a quadratic equation is a polynomial whose highest degree is 2. Let us see some examples:

• 3x²+x+1, where a=3, b=1, c=1
• 9x²-11x+5, where a=9, b=-11, c=5

Roots of Quadratic Equations:

If we solve any quadratic equation, then the value we obtain are called the roots of the equation. Since the degree of the quadratic equation is two, therefore we get here two solutions and hence two roots.

There are different methods to find the roots of quadratic equation, such as:

• Factorisation
• Completing the square
• Using quadratic formula

Learn: Factorization of Quadratic equations

Quadratic Equation Formula:

The quadratic formula to find the roots of the quadratic equation is given by:

Where b²-4ac is called the discriminant of the equation.

Based on the discriminant value, there are three possible conditions, which defines the nature of roots as follows:

• two distinct real roots, if b² – 4ac > 0
• two equal real roots, if b² – 4ac = 0
• no real roots, if b² – 4ac < 0

Also, learn quadratic equations for class 10 here.

Quadratic Equations Problems and Solutions

1. Rahul and Rohan have 45 marbles together. After losing 5 marbles each, the product of the number of marbles they both have now is 124. How to find out how many marbles they had to start with.

Solution: Say, the number of marbles Rahul had be x.

Then the number of marbles Rohan had = 45 – x.

The number of marbles left with Rahul after losing 5 marbles = x – 5

The number of marbles left with Rohan after losing 5 marbles = 45 – x – 5 = 40 – x

The product of number of marbles = 124

(x – 5) (40 – x) = 124

40x – x² – 200 + 5x = 124

– x² + 45x – 200 = 124

x² – 45x + 324 = 0

This represents the quadratic equation. Hence by solving the given equation for x, we get;

x = 36 and x = 9

So, the number of marbles Rahul had is 36 and Rohan had is 9 or vice versa.

2. Check if x(x + 1) + 8 = (x + 2) (x – 2) is in the form of quadratic equation.

Solution: Given,

x(x + 1) + 8 = (x + 2) (x – 2)

x²+x+8 = x²-2² [By algebraic identities]

Cancel x² both the sides.

x+8=-4

x+12=0

Since, this expression is not in the form of ax²+bx+c, hence it is not a quadratic equation.

3. Find the roots of the equation 2x² – 5x + 3 = 0 using factorisation.

Solution: Given,

2x² – 5x + 3 = 0

2x² – 2x-3x+3 = 0

2x(x-1)-3(x-1) = 0

(2x-3) (x-1) = 0

So,

2x-3 = 0; x = 3/2

(x-1) = 0; x=1

Therefore, 3/2 and 1 are the roots of the given equation.

4. Solve the quadratic equation 2x²+ x – 300 = 0 using factorisation.

Solution: 2x²+ x – 300 = 0

2x² – 24x + 25x – 300 = 0

2x (x – 12) + 25 (x – 12) = 0

(x – 12)(2x + 25) = 0

So,

x-12=0; x=12

(2x+25) = 0; x=-25/2 = -12.5

Therefore, 12 and -12.5 are two roots of the given equation.

Also, read Factorisation.

5. Solve the equation x²+4x-5=0.

Solution:

x² + 4x – 5 = 0

x²-1x+5x-5 = 0

x(x-1)+5(x-1) =0

(x-1)(x+5) =0

Hence, (x-1) =0, and (x+5) =0

x-1 = 0

x = 1

similarly, x+5 = 0

x =-5.

Therefore,

x=-5 & x=1

6. Solve the quadratic equation 2x² + x – 528 = 0, using quadratic formula.

Solution: If we compare it with standard equation, ax²+bx+c = 0

a=2, b=1 and c=-528

Hence, by using the quadratic formula:

Now putting the values of a,b and c.

x=64/4 or x=-66/4

x=16 or x=-33/2

7. Find the roots of x² + 4x + 5 = 0, if any exist, using quadratic formula.

Solution: To check whether there are real roots available for the quadratic equation, we need the find the discriminant value.

D = b²-4ac = 4²– 4(1)(5) = 16-20 = -4

Since the square root of -4 will not give a real number. Hence there is no real roots for the given equation.

8. Find the discriminant of the equation: 3x²-2x+⅓ = 0.

Solution: Here, a = 3, b=-2 and c=⅓

Hence, discriminant, D = b² – 4ac

D = (-2)²-4(3)(⅓)

D = 4-4

D=0

Video Lesson

Quadratic Equation Worksheet

Practice Questions

Solve these quadratic equations and find the roots. 

1. x²-5x-14=0         [Answer: x=-2 & x=7]
2. X² = 11x -28       [Answer: x=4 & x = 7]
3. 6x²– x = 5            [Answer: x=-⅚ & x = 1]
4. 12x² = 25x          [Answer: x=0 & x=25/12]

Frequently Asked Questions on Quadratic Equations

Q1

What is a quadratic equation?

A polynomial which has the highest degree equal to two is a quadratic equation which is expressed in the form of ax²+bx+c=0, where x is the variable and a,b,c are the real numbers & a ≠ 0.

Q2

What are the examples of quadratic equations?

The examples of quadratic equations are:
5x² – x + 6 = 0
x² + 8x + 2 = 0
-x² + 6x + 18 = 0
x² – 4 = 0

Q3

What is the formula for quadratics?

The formula to find the solution of quadratic equations is:
x = [-b ± √(b²-4ac)]/2a

Q4

What are the methods to solve the quadratic equation?

There are basically four methods to find the solution of quadratic equations.
Factorisation
Square root property
Completing the square
Using the quadratic formula

Q5

What are the roots of the quadratic equation?

Roots of the quadratic equation are nothing but the solution for a two-degree polynomial. Since the degree of a quadratic equation is two, therefore, there will be two roots or two solutions for the equation. The formula to find the two roots of equation ax²+bx+c = 0, are:
x1 = [-b + √(b²-4ac)]/2a and
x2 = [-b – √(b²-4ac)]/2a

Q6

What are the zeroes of the quadratic equation?

The zeroes of the quadratic equation are those values for which the polynomial of the equation (i.e. left-hand side of ax²+bx+c = 0) is equal to zero. They are also called roots of the equation.