CBSE Class 10 Mathematics Quadratic Equations Notes Set 10

Quadratic Equations

A quadratic equation has the form \( ax^2+bx+c=0 \) with a not zero. If a = 0 it becomes linear. It has exactly two roots, real or not real. Special types: b = 0 (\( ax^2+c=0 \)), c = 0 (\( ax^2+bx=0 \)) and b = c = 0.
The roots are the values of x that satisfy the equation, equivalently the x-values where the parabola \( y=ax^2+bx+c \) meets the x-axis.

Methods of Solving

Factorisation: find two numbers whose product is ac and whose sum is b, split the middle term, group and set each factor to zero. Completing the square: divide by a, move the constant across, add \( \left(\frac{b}{2a}\right)^2 \) to both sides and take square roots. Quadratic formula: \( x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} \).
Equations with the unknown in denominators must be cleared of fractions first, and any value that makes a denominator zero is rejected.

Discriminant and Nature of Roots

Let \( D=b^2-4ac \). If D > 0 the roots are real and distinct (rational when D is a perfect square, otherwise irrational). If D = 0 the roots are real and equal, each \( -\frac{b}{2a} \). If D < 0 there are no real roots.
Sum of roots \( =-\frac{b}{a} \) and product of roots \( =\frac{c}{a} \). Reciprocal roots need c = a. For equal roots, put D = 0; for a perfect square expression, also put D = 0.

Word Problems

Name the unknown, form the equation from the statement and solve it. Then discard roots that do not make sense, such as a negative length, age or number of objects. Typical topics: consecutive integers (x, x + 1), areas of rectangles, right triangles (Pythagoras), speed with a stream (upstream 5 - x, downstream 5 + x), work done by two taps, ages, and digits of a number.
Example: sides of a right triangle with hypotenuse 25 cm differing by 5 cm satisfy \( x^2+(x+5)^2=625 \), giving the sides 15 cm and 20 cm.