# Thread: Expansion and Factorisation of Algebraic Expressions

1. ## Expansion and Factorisation of Algebraic Expressions

1. If a + b = x and ab = y, express (a − b)^2 in terms of x and y.

(A) x^2 − y (B) x2 ^− 2y (C) x^2 − 4y (D) x^2 + 2y (E) x^2 − y^2

if i apply the formula for (a − b)^2 then i dont get any of the answers.the problem comes as to where a+b=x should be used.

2. Simplify (−a − b) 2 − (a + b)(a − b) .

-(a^2+2ab+b^2)-a^2-b^2=-2a^2-2ab=-2a(a-b)

But the actual answer is 2b^2 + 2ab.How ?

2. Originally Posted by haftakhan
1. If a + b = x and ab = y, express (a − b)^2 in terms of x and y.

(A) x^2 − y (B) x2 ^− 2y (C) x^2 − 4y (D) x^2 + 2y (E) x^2 − y^2

if i apply the formula for (a − b)^2 then i dont get any of the answers.the problem comes as to where a+b=x should be used.
$\displaystyle a + b = x \implies (a + b)^2 = x^2 \implies a^2 + 2ab + b^2 = x^2 \implies a^2 + 2y + b^2 = x^2$

So from the first eqaution, we get $\displaystyle a^2 + b^2 = x^2 - 2y$

Hence, $\displaystyle (a - b)^2 = a^2 - 2ab + b^2 = x^2 - 2y - 2y = x^2 - 4y$

2. Simplify (−a − b) 2 − (a + b)(a − b) .

-(a^2+2ab+b^2)-a^2-b^2=-2a^2-2ab=-2a(a-b)

But the actual answer is 2b^2 + 2ab.How ?
$\displaystyle (-a - b)^2 - (a + b)(a - b) = (a + b)^2 - (a^2 - b^2)$

$\displaystyle = a^2 + 2ab + b^2 - a^2 + b^2$

$\displaystyle = 2ab + 2b^2$

the minus sign in front of $a^2 + 2ab + b^2$was your problem. it is not the same as what you had before. squaring gets rid of the negative

3. But where has the minus at the start has gone ?

4. Originally Posted by haftakhan
But where has the minus at the start has gone ?
$\displaystyle (-a - b)^2 = [-(a + b)]^2 = (-1)^2 (a + b)^2 = (a + b)^2$

5. hmm.Now i get it.
Thanks for all the help