Quadratic Equations
Quadratic Equations – Equations in which the variable is raised to the highest power of two are called quadratic equations.
ax2 + bx + c = 0
Sridharacharya Sutra / Quadratic Formula –
Nature of Roots –
1. Roots are real and different
2. Roots are real and equal
3. Roots are imaginary
Que. 1. Check whether the following are quadratic equations:
1. (x + 1)2 = 2 (x – 3)
[ (a + b)2 = a2 + 2ab + b2 ]
X2 + 1 + 2x 1 = 2x – 6
X2 + 1 + 2x = 2x – 6
X2 + 2x – 2x + 1 + 6 = 0
X2 + 7 = 0
This is a quadratic equation.
2. X2 – 2x = ( -2 ) (3 – x )
X2 – 2x = -6 + 2x
X2 – 2x – 2x + 6 = 0
X2 – 4x + 6 = 0
This is a quadratic equation.
3. (x – 2) (x + 1) = (x – 1) (x + 3)
X2 + x – 2x – 2 = x2 + 3x -x – 3
X2 – x – 2 = x2 + 2x – 3
X2 – X2 – x – 2x – 2 + 3 = 0
-3x + 1 = 0
This is not a quadratic equation.
4. (x – 3) (2x + 1) = x (x + 5)
2x2 – 5x – 3 = x2 + 5x
2x2 – 5x – 3 – x2 – 5x = 0
2x2 – x2 – 5x – 5x – 3 = 0
X2 – 10x – 3 = 0
This is a quadratic equation.
5. (2x – 1) (x – 3) = (x + 5) (x – 1)
2x2 – 6x – x + 3 = x2 – x + 5x – 5
2x2 – 7x + 3 = x2 + 4x – 5
2x2 – x2 – 7x – 4x + 3 + 5 = 0
X2 – 11x + 8 = 0
This is a quadratic equation.
6. x2 + 3x + 1 = (x – 2)2
x2 + 3x + 1 = x2 – 4x + 4
x2 – x2 + 3x + 4x + 1 – 4 = 0
7x – 3 = 0
This is not a quadratic equation.
7. (x + 2)3 = 2x (x2 – 1)
[ (x + y)3 = x3 + y3+ 3xy (x + y) ]
= x3+ y3+ 3x2y + 3xy2
X3 + ( 2 )3 + 3 ( x )2 X 2 + 3x ( 2 )2 = 2x3 – 2x
X3 + 8 + 6x2 + 12x = 2x3 – 2x
X3 – 2x3 + 6x2 + 12x + 2x + 8 = 0
-x3 + 6x2 + 14x + 8 = 0
This is not a quadratic equation.
8. x3 – 4x2 – x + 1 = (x – 2)3
x3 – 4x2 – x + 1 = x3 – ( 2 )3 – 3 ( x )2 X 2 + 3x (-2)2
[ (x + y)3 = x3 – y3 – 3x2y + 3xy2 ]
X3 – 4x2 – x + 1 = x3 – 8 – 6x2 + 12x
X3 – x3 – 4x2 + 6x2 – x – 12x + 1 + 8 = 0
2x2 – 13x + 9 = 0
This is a quadratic equation.
2. Represent the following situations as quadratic equations:
(i) The area of a rectangular plot is 528 m. The length (in meters) of the plot is one more than twice its width. We need to find the length and width of the plot.
Let the width of the rectangular plot = x
Let the length of the rectangular plot = 2x + 1
As per the question,
Length x width = 528 m2
(2x + 1) X x = 528
2x2 + x = 528
2x2 + x – 528 = 0
(ii) The product of two consecutive positive integers is 306. We have to find the integers.
Let the first positive integer = x
Let the second positive integer = x + 1
As per the question,
X (x + 1) = 306
X2 + x = 306
X2 + x – 306 = 0
(iii) Rohan’s mother is 26 years older than him. Three years from now, the product of their ages (in years) will be 360. We need to find Rohan’s present age.
Let Rohan’s age = x years
Let Rohan’s mother’s age = (x + 26) years
After three years,
Rohan’s age = x + 3
Rohan’s mother’s age = x + 26 + 3
= x + 29
(x + 3) (x + 29) = 360
X2 + 29x + 3x +87 = 360
X2 + 32x + 87 – 360 = 0
X2 + 32x – 273 = 0
(iv) A train travels a distance of 480 km at a uniform speed. If its speed were 8 km/h less, it would take 3 hours more to cover the same distance. We need to find the speed of the train.
Let the speed of the train = x
Let the distance of the train = 480 km
Time = t
Speed = (x – 8) km/h
Distance = 480 km
Time = (t + 3) hour
(x – 8) (480 + 3x) = 480x
480x + 3x2 – 3840 – 24x = 480x
480x + 3x2 – 3840 – 24x – 480x = 0
3x2 – 24x – 3840 = 0
