Question

factorise

$a{2} + a - 42$
pls explain​

Answer 1

To Factorise (Appropriate) :

$\tt{ {a}^{2} + a - 42}$

Solution :

We will use the middle term splitting method ,

The factors of -42 can be :

• -42 = -6×7
• -42 = -7×6
• -42 = -21×2
• -42 = -14×3
• -42 = -3×14
• -42 = -2×21 and etc.

But, only the pair (-6,7) fits best for spliting

So, a²+a-42

= a²+7a-6a-42

= a(a+7)-6(a+7)

= (a+7)(a-6)

Hence, The Solution is :

a²+a-42 = (a+7)(a-6)

Additional Information :

• (a+b)(a-b) = a²-b²
• (a+b)² = a²+b²+2ab
• (a-b)² = a²+b²-2ab
• (x+a)(x+b) = x²+(a+b)x+ab
• (a+b+c)² = a²+b²+c²+2ab+2bc+2ac
• (a+b)³ = a³+b³+3a²b+3ab²
• (a-b)² = a²-b³-3a²b+3ab²

Answer 2

$\large\underline{\text{How to factorize? (A.C method)}}$

Quadratic polynomials: $ax^{2}+bx+c,\ a\neq0$

The factorization is the opposite method to expansion. The factorization shows the polynomial is a product of which factors.

The A.C method, named from the leading coefficient $a$ and the lowest degree coefficient $c$, can factorize quadratic polynomials.

Let's list the pairs of numbers when multiplied gives -42.

List of pairs:-

• $\pm1,\mp42$
• $\pm2,\mp21$
• $\pm3,\mp14$
• $\pm6,\mp7$

(Here, $\pm$ is the double sign and $\mp$ stands for the opposite sign.)

Let's find the sum of each number pair.

• $\pm(1-42)=\mp41$
• $\pm(2-21)=\mp19$
• $\pm(3-14)=\mp11$
• $\pm(6-7)=\mp1$

Now, we can see that the only pair with the sum is 7 and -6.

$\therefore a^{2}+a-42=(a+7)(a-6)$