Mathematics class 12th

1. QUADRATIC EQUATION
CLASS 10TH

2. OBJECTIVES
 QUADRATIC EQUATION DEFINITION
 METHODS TO FIND THE SOLUTION OF
QUADRATIC EQUATIONS
 DISCRIMINANT
 NATURE OF ROOTS
 EXAMAPLES

3. QUADRATIC EQUATION
A polynomial p(x) = ax2 + bx + c of degree 2 is called a
quadratic polynomial, then p(x) = 0 is known as a quadratic
equation.
e.g. 2x2 – 3x + 2 = 0, x2 + 5x + 6 = 0 are quadratic equations.

4. METHODS TO FIND THE SOLUTION OF
QUADRATIC EQUATIONS
Three methods to find the solution of a quadratic equation:
1.Factorization method
2. Method of completing the square
3. Quadratic formula method


5. FACTORISATION METHOD
Steps to find the solution of a given quadratic equation by factorization:
Firstly, write the given quadratic equation in standard form ax2 + bx + c = 0.
Find two numbers α and β such that the sum of α and β is equal to b and the product
of α and β is equal to ac.
Write the middle term bx as αx + β x and factorize 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 the reach factor to zero and find the values of x.
These values of x are the required roots/solutions of the given quadratic
equation.

6. Find the roots of the quadratic equation 6x2 – x – 2 = 0
Solution : We have 6x2 – x – 2 = 6x2 + 3x – 4x – 2
= x (2x + 1) – 2 (2x + 1)
= (3x – 2)(2x + 1)
The roots of 6x2 – x – 2 = 0 are the values of x for which (3x – 2)(2x + 1) = 0
Therefore, 3x – 2 = 0 or 2x + 1 = 0,
i.e., x = 2/3 or x = -1/2
Therefore, the roots of 6x2 – x – 2 = 0 are 2/3 and –1/2
We verify the roots, by checking that 2/3 and -1/2 satisfy 6x2 – x – 2 = 0

7. METHOD OF COMPLETING THE SQUARE
Steps for Completing the Square Method:
Suppose ax2 + bx + c = 0 is the given quadratic equation. Then follow
the given steps to solve it by completing the square method.
 Step 1: Write the equation in the form, such that c is on the right
side.
 Step 2: If a is not equal to 1, divide the complete equation by a
such that the coefficient of x2 will be 1.
 Step 3: Now add the square of half of the coefficient of term-x,
(b/2a)2, on both sides.
 Step 4: Factorize the left side of the equation as the square of the
binomial term.
 Step 5: Take the square root on both sides

8. The above steps can be implemented as shown below.
Consider the quadratic equation ax2 + bx + c = 0 (a ≠ 0).
Dividing throughout by a, we get
x2 + (b/a)x + (c/a) = 0
This can also be written by adding and subtracting (b/2a)2 as,
[x + (b/2a)]2
– (b/2a)2
+ (c/a) = 0
[x + (b/2a)]2
– [(b2 – 4ac)/4a2] = 0
[x + (b/2a)]2
= [(b2 – 4ac)/4a2]
If b2 – 4ac ≥ 0, then by taking the square root, we get
x + (b/2a) = ± √(b2 – 4ac)/ 2a
Further simplification of this will give you the quadratic
formula.

9. Example: Solve 4x2 + x = 3 by completing the square
method.
Solution: Given,
4x2 + x = 3
Divide the whole equation by 4.
x2 + x/4 = ¾
Coefficient of x is ¼
Half of ¼ = ⅛
Square of ⅛ = (⅛)
2
Add (⅛)2 on both sides of the equation.
x2 + x/4 + (⅛)2 = ¾ – (⅛)2
x2 + x/4 + 1/64 = ¾ + 1/64
(x + ⅛)
2
= (48+1)/64 = 49/64
Taking square root on both sides, we get;
x+1/8 = √(49/64) = ±7/8
x = ⅞ – ⅛ = 6/8 = ⅜
x = – ⅞ – ⅛ = -8/8 = -1

10. QUADRATIC FORMULA METHOD
If b2 – 4ac ≥ 0, then the roots of the quadratic equation
ax2 + bx + c = 0 are given by
x = [-b ± √(b2 –
4ac)]/ 2a
This formula for finding the roots of a quadratic equation
is known as the quadratic formula.

11. Solve x2 – 5x + 6 = 0.
Solution: Given,
x2 – 5x + 6 = 0
a = 1, b = -5, c = 6
By the quadratic formula, we know;
x = [-b ± √(b2 – 4ac)]/ 2a
b2 – 4ac = (-5)2 – 4 × 1 × 6 = 25 – 24 = 1 > 0
Thus, the roots are real.
Hence,
x = [-b ± √(b2 – 4ac)]/ 2a
= [-(-5) ± √1]/ 2(1)
= [5 ± 1]/ 2
i.e. x = (5 + 1)/2 and x = (5 – 1)/2
x = 6/2, x = 4/2
x = 3, 2
Therefore, the solution of x2 – 5x + 6 = 0 is 3 or 2.

12. DISCRIMINANT
 The quadratic equation ax2 + bx + c = 0 has real roots or not,
b2 – 4ac is called the discriminant of this quadratic equation.

13. Determine the discriminant value for the given quadratic
equation 2x2+8x+8.
Solution:
Given: The quadratic equation is 2x2+8x+8
Here, the coefficients are:
a = 2, b = 8 and c = 8
The formula to find the discriminant value is D = b2 –
4ac
Now, substitute the values in the formula
Discriminant, D = 82 – 4(2)(8)
D = 64 – 4 (16)
D = 64 – 64
D = 0
The discriminant value is 0

14. NATURE OF ROOTS
A quadratic equation ax2 + bx + c = 0 has
(i) Two distinct real roots, if b2 – 4ac > 0,
(ii) Two equal real roots, if b2 – 4ac = 0,
(iii) No real roots, if b2 – 4ac < 0.

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

16. THE END