Question

Factorise by splitting the middle term.
$3 {x}^{2} + 60xy - 288 {y}^{2}$
​

Answer 1

Answer:

3(x + 24y)(x - 4y)

Step-by-step explanation:

Given Equation:

$3x^2 + 60xy - 288y^2$

From the given equation, take 3 out common.

We get

$3(x^2 + 20xy - 96y^2)$

Now, take only the middle part of the equation.

$x^2 + 20xy - 96y^2$

To split the middle term correctly, we must make sure that the sum of the 2 new terms that are formed equals to the product of the first and the last term of the equation.

Let me show you how.

Note:

The signs preceding the terms are very very important. Messing up the signs messes up your answer.

Product of the 1st & last term = 1 × (-96)

                                                = -96

To split the middle term, i.e., 20xy, the sum of 2 terms should be equal to -96. You have to use the trial and error method, i.e., by trying out random numbers.

Let's try it out now.

Factors of -96 are:

1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96, -1,- 2, -3, -4, -6, -8,- 12, -16, -24, -32, -48, -96.

To get -96 as the product, we can use 1 negative and 1 positive number.

After trial and error, we can split 20xy as 24xy and -4xy. [24xy + (-4xy)

                                                                                                 = 24xy - 4xy

                                                                                                 = 20xy]

Now, the equation can be rewritten as

$3(x^2 + 24xy - 4xy - 94y^2)$

Take the first 2 terms and the last 2 terms together, take the middle term of each part out, and you will get the answer.

3[x(x + 24y) - 4y(x + 24y)]

⇒ 3(x + 24y)(x - 4y)