class 10

CONTENTS OF THIS PAGE:

Quadratic Equations

Factorisation

Steps involved in factorisation

Splitting the middle term

Synthetic division (video)

Quadratic formula (video)

Completing the square method (video)

QUADRATIC EQUATIONS:

Quadratic equation is an equation of the form ax2 + bx + c = 0 where x is a variable and a, b, c are real numbers and a≠0. In other words, it a polynomial of degree 2.

Examples: x2-6x+9=0

5y2+11y+6

Factorisation:

It is a method in which an algebraic expression is written as a product of its factors. In a quadratic polynomial, firstly we see if there is any common factor for all the terms. If so, we take the common factor out and then factorise the expression to the extent that there can be no further simplification. The final answer will be a product of the common factor and all other algebraic factors.

Steps involved to factorise ax2 + bx + c:

Write the values of a, b and c from the given expression
Find the product of a and c.
Choose two factors of ac such that their product is ac and their sum is b
In place of b, write these two factors along with the variable present in the sum.
This is called splitting the middle term. Here b is the middle term.
From the first two terms, take the common factor out.
From the next two terms, take the common factor out.
See to it that the remaining terms inside the two brackets are one and the same.
The final answer will be the product of the one inside the bracket and the other expression (which is the sum of the common factors already taken out)

Let us see some examples:

Factorise

x2+7x+12

= x2+4x+3x+12

= x(x+4) + 3(x+4)

= (x+4) (x+3)

Here, we have chosen two factors of 12 such that their product gives 12 and their sum gives 7.

Factorise

x2-3x-40

= x2-8x+5x+40

= x(x-8) + 5(x-8)

= (x+5) (x-8)

Here, we choose two factors of 40 such that their product is -40 and their sum is -3.

Factorise

10x2-2x-12

= 2(5x2-x-6)

= 2(5x2+5x-6x-6)

= 2(5x(x+1)-6(x+1))

= 2 (x+1) (5x-6)

Here, we take 2 as the common factor and write the reduced terms inside the bracket. Then we follow the method of splitting the middle term explained above.

SYNTHETIC DIVISION OF CUBIC POLYNOMIALS

METHOD OF COMPLETING THE SQUARE

QUADRATIC FORMULA

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