Question

if a-b=4and a+b=6find, ab​

Answer 1

Given,

a + b = 6 …… (1)

a - b = 4 ……. (2)

Method 1: (easiest)

add equation (1) and (2), we get

2a = 6 + 4

=> 2a = 10

=> a = 5

substitute value of a in either of equation (1) or (2),

=> b = 1

Hence ab = 5.1 = 5

————-

Method 2:

we know

(a+b)ˆ2 = aˆ2 + bˆ2 + 2ab

(a-b)ˆ2 = aˆ2 + bˆ2 - 2ab

subtracting equation 2 from 1, we get

(a+b)ˆ2−(a−b)ˆ2=(aˆ2+bˆ2+2ab)−(aˆ2+bˆ2−2ab)

=>(a+b)ˆ2−(a−b)ˆ2=aˆ2+bˆ2+2ab−(aˆ2+bˆ2−2ab)

=>(a+b)ˆ2−(a−b)ˆ2=aˆ2+bˆ2+2ab−aˆ2−bˆ2+2ab

=>(a+b)ˆ2−(a−b)ˆ2=4ab ………… (3)

substituting the values of a+b and a-b in the equation (3) above, we have

=>6ˆ2−4ˆ2=4ab

=>36−16=4ab

=>20=4ab

=>4.5=4ab

=>ab=5

Hence ab = 5

Answer 2

Answer:

$(a - b) + (a + b) =( 4 + 6) \\ (a - b) + (a + b) = 10 \\ so \:( - b)and( + b) \: cancel \\ it \: left \: with \: \\ (+ a) \: and \: ( + a) \\ so \: it \: can \: be \: written \: as \: 2a = 10 \\ a = \frac{10}{2} = 5 \\ a = 5 \: \\ so \: a + b = 10 \\ 5 + b = 10 \\ b = 10 - 5 = 5 \\ so \: ab = 5 \times 5 = 25$