1. Check whether the following are quadratic equations :
i. (x + 1)2
= 2(x – 3)
ii. x
2
– 2x = (–2) (3 – x)
iii. (x – 2)(x + 1) = (x – 1)(x + 3)
iv. (x – 3)(2x +1) = x(x + 5)
v. (2x – 1)(x – 3) = (x + 5)(x – 1)
vi. x
2
+ 3x + 1 = (x – 2)2
vii. (x + 2)3
= 2x (x
2
– 1)
viii. x
3
– 4x
2
– x + 1 = (x – 2)3
2. Represent the following situations in the form of quadratic equations :
i. The area of a rectangular plot is 528 m2
.
The length of the plot is one metre more than
twice its breadth.
We need to find the length and breadth of the plot.
ii. The product of two consecutive positive integers is 306.
We need to find the integers.
iii. Rohan’s mother is 26 years older than him.
The product of their ages after 3 years will
be 360 years. We need to find Rohan’s present age.
iv. A train travels a distance of 480 km at a uniform speed. If the speed had been 8 km/h
less, then it would have taken 3 hours more to cover the same distance. We need to
find the speed of the train.
5.3 SOLUTION OF A QUADRATIC EQUATION BY FACTORISATION
We have learned to represent some of the daily life situations in the form of quadratic
equation with an unknown variable ‘x’.
Now we need to find the value of x.
Consider the quadratic equation 2x
2
– 3x + 1 = 0. If we replace x by 1. Then, we get
(2 × 12
) – (3 × 1) + 1 = 0 = RHS of the equation. Since 1 satisfies the equation , we say that 1
is a root of the quadratic equation 2x
2
– 3x + 1 = 0.
\ x = 1 is a solution of the quadratic equation.
This also means that 1 is a zero of the quadratic polynomial 2x
2
– 3x + 1.
In general, a real number a is called a root of the quadratic equation ax
2
+ bx + c = 0,
if aa
2
+ b a + c = 0.
We also say that x = a is a solution of the quadratic equation, or
a satisfies the quadratic equation.
Note that the zeroes of the quadratic polynomial ax
2
+ bx + c (a ¹ 0) and the roots
of the quadratic equation ax
2
+ bx + c = 0 (a ¹ 0) are the same.
We have observed, in Chapter 3, that a quadratic polynomial can have at most two zeroes.
So, any quadratic equation can have at most two roots. (Why?)