Quadratic equations Most important questions

CHAPTER – 4
QUADRATIC EQUATIONS
FACTORISATION METHODS TO FIND THE SOLUTION OF QUADRATIC EQUATIONS
Steps to find the solution of given quadratic equation by factorisation
 Firstly, write the given quadratic equation in standard form ax2 + bx + c = 0.
 Find two numbers  and  such that sum of  and  is equal to b and product of  and  is
equal to ac.
 Write the middle term bx as  x   x and factorise it by splitting the middle term and let factors
are (x + p) and (x + q) i.e. ax2 + bx + c = 0  (x + p)(x + q) = 0
 Now equate reach factor to zero and find the values of x.
 These values of x are the required roots/solutions of the given quadratic equation.

IMPORTANT QUESTIONS
Solve the quadratic equation by using factorization method: x2 + 2x – 8 = 0
Solution: x2 + 2x – 8 = 0
 x2 + 4x – 2x – 8 = 0  x(x + 4) – 2(x + 4) = 0
 (x + 4)(x – 2) = 0  x + 4 = 0, x – 2 = 0  x = – 4, 2

Questions for practice
1. Solve the quadratic equation using factorization method: x2 + 7x – 18 = 0
2. Solve the quadratic equation using factorization method: x2 + 5x – 6 = 0
3. Solve the quadratic equation using factorization method: y2 – 4y + 3 = 0
4. Solve the quadratic equation using factorization method: x2 – 21x + 108 = 0
5. Solve the quadratic equation using factorization method: x2 – 11x – 80 = 0
6. Solve the quadratic equation using factorization method: x2 – x – 156 = 0
1 1 1 1
7. Solve the following for x :    .
ab x a b x
1 1 1 1
8. Solve the following for x :   
2a  b  2 x 2a b 2 x

NATURE OF ROOTS
The roots of the quadratic equation ax2 + bx + c = 0 by quadratic formula are given by
b  b 2  4ac b  D
x 
2a 2a
2
where D = b  4ac is called discriminant. The nature of roots depends upon the value of
discriminant D. There are three cases –
Case – I
When D > 0 i.e. b 2  4ac > 0, then the quadratic equation has two distinct roots.
b  D b  D
i.e. x  and
2a 2a
Case – II
When D = 0, then the quadratic equation has two equal real roots.
b b
i.e. x  and
2a 2a
Case – III
When D < 0 then there is no real roots exist.

, IMPORTANT QUESTIONS
Find the discriminant of the quadratic equation 2x2 – 4x + 3 = 0, and hence find the nature of
its roots.
Solution : The given equation is of the form ax2 + bx + c = 0, where a = 2, b = – 4 and c = 3.
Therefore, the discriminant, D = b2 – 4ac = (– 4)2 – (4 × 2 × 3) = 16 – 24 = – 8 < 0
So, the given equation has no real roots.

Questions for Practice
1. Find the discriminant and the nature of the roots of quadratic equation: 3 3 x2 + 10x + 3 = 0.
2. Find discriminant and the nature of the roots of quadratic equation: 4x2 – 2x2 + 3 = 0.
3. Find discriminant and the nature of the roots of quadratic equation: 4x2 – 12x + 9 = 0.
4. Find discriminant and the nature of the roots of quadratic equation: 5x2 + 5x + 6 = 0.
5. Write the nature of roots of quadratic equation 4x2 + 4 3 x + 3 = 0.
6. Write the nature of roots of the quadratic equation 9x2 – 6x – 2 = 0.
7. Write the nature of roots of quadratic equation : 4x2 + 6x + 3 = 0
8. The roots of ax2 + bx + c = 0, a ≠ 0 are real and unequal. What is value of D?
9. If ax2 + bx + c = 0 has equal roots, what is the value of c?

QUADRATIC FORMULA METHOD
Steps to find the solution of given quadratic equation by quadratic formula method:
 Firstly, write the given quadratic equation in standard form ax2 + bx + c = 0.
 Write the values of a, b and c by comparing the given equation with standard form.
 Find discriminant D = b2 – 4ac. If value of D is negative, then is no real solution i.e. solution
does not exist. If value of D  0, then solution exists follow the next step.
b  D
 Put the value of a, b and D in quadratic formula x  and get the required
2a
roots/solutions.

IMPORTANT QUESTIONS
Solve the quadratic equation by using quadratic formula: x2 + x – 6 = 0
Solution: Here, a = 1, b = 1, c = –6
 D = b2 – 4ac = 1 – 4(1)( –6) = 1 + 24 = 25 > 0
b  D 1  25 1  5 1  5 1  5 6 4
Now, x    x or x or  x  3 or 2
2a 2(1) 2 2 2 2 2
Questions for practice
1. Solve the quadratic equation by using quadratic formula: x2 – 7x + 18 = 0
2. Solve the quadratic equation by using quadratic formula: x2 – 5x + 6 = 0
3. Solve the quadratic equation by using quadratic formula: y2 + 4y + 3 = 0
4. Solve the quadratic equation by using quadratic formula: x2 + 11x – 80 = 0
5. Solve the quadratic equation by using quadratic formula: x2 + x – 156 = 0
6. Solve for x by using quadratic formula : 9x2 – 9(a + b)x + (2a2 + 5ab + 2b2) = 0.