Revision Notes: Quadratic Equations | Mathematics Class 10 ICSE PDF Download

Important Concepts

1. If p(x) is a quadratic polynomial, then p(x) = 0 is called a quadratic equation. 
   The general form of a quadratic equation, in the variable x, is ax² + bx + c = 0, where a, b, c are real numbers and a ≠ 0.
2. The value of x that satisfies an equation is called the zeroes or roots of the equation. 
   A real number α is said to be a solution/root of the quadratic equation ax² + bx + c = 0
   if aα² + bα + c = 0.
3. A quadratic equation has at most two zeros.

Nature of the roots of a quadratic equation

The nature of the roots of a quadratic equation depends upon the value of discriminant b² – 4ac.
i. If b² – 4ac > 0, the roots are real and unequal
ii. If b² – 4ac = 0, the roots are real and equal
iii. If b² – 4ac < 0, the roots are imaginary (not real) 
If ax² + bx + c, a ≠ 0, can be reduced to the product of two linear factors, then the roots of the quadratic equation ax² + bx + c = 0 can be found by equating each factor to zero.

Solution of Quadratic Equation by Factroisation

Example, Solve the equation (9/2)x = 5 + x² by factorization: 
Step 1: Clear all fractions and brackets, if necessary
9x = 2(5 + x²)
Step 2: Transpose all the terms to the left hand side to get an equation in the form ax² + bx + c = 0
9x = 2x² + 10
⇒ 2x² - 9x +10 = 0
Step 3: Factorise the expression on the left hand side.
2x² - 9x + 10 = 0
⇒ 2x²- 5x - 4x +10 = 0
⇒ x (2x - 5) -2 (2x - 5) = 0
⇒ (x - 2)(2x - 5) = 0
Step 4: Put each factor equal to zero and solve
(x - 2) (2x - 5) = 0
⇒ x - 2 = 0, (2x - 5) = 0
⇒ x = 2; 2x = 5
⇒ x = 2;  x = 5/2
Thus, we have, x = 2 or x = 5/2

Solution of Quadratic Equation by Quadratic Formula

The roots of a quadratic equation ax²+ bx + c = 0 (a ≠ 0) can be calculated by using quadratic formula:

where b² - 4ac ≥ 0

Equations Reducible to Quadratic Form

There are many equations which are not in the quadratic form but can be reduced to quadratic form by simplifications.
Let us solve the equation x⁴ - 2x² - 3 = 0
It is clear that the above equation is not a quadratic equation.
Now assume that x² = y
Then rewrite the given quadratic equation as,

Substituting in the above equation, we have y² - 2y - 3 = 0
This is a quadratic equation in y.
Let us solve the quadratic equation through factorization.
y² - 2y - 3 = 0
⇒ y² - 3y + y - 3 = 0
⇒ y (y - 3) +(y - 3) = 0
⇒ (y + 1)(y - 3) = 0
⇒ y + 1 = 0 or y - 3 = 0
⇒ y = -1 or y = 3

Applications of quadratic equation in solving real life problems

Following points can be helpful in solving word problems:
i. Every two digit number ‘xy’ where x is a ten’s place and y is a unit’s place can be expressed as 
xy = 10x + y
ii. Downstream: It means that the boat is running in the direction of the stream 
Upstream: It means that the boat is running in the opposite direction of the stream 
Thus, if
Speed of boat in still water is x km/h
And the speed of stream is y km/h
Then the speed of boat downstream will be (x + y) km/h and in upstream it will be (x − y) km/h.
iii. If a person takes x days to finish a work, then his one day's work = 1/x