Question

please explain me how they have done first two sums.

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I don't want solution i want its explanation​

Answer 1

It is done by the method of Factoring the trinomials by breaking the middle term of the equation into real numbers for example ax²+bx+c such that the algebraic sum of these two numbers is b and their product is ac, then factorise by grouping method.

Answer 2

Well, first let me tell you how factorization method works.

Factorization method :

In this method, target is to divide the middle term of equation in two parts such that :

• Product of two parts' constant term = Product of constant term of 1st term & 3rd term.
• Sum of 2 parts = Middle term.

So using this method let's solve first equation :.

Solution ❶

First equation : 4√3x² + 5x - 2√3

Now we will divide the middle term in two such parts as told above.

At first, product of 1st term & 3rd term:

⇒ 4√3x² × -2√3

⇒ 8(√3)² x²

⇒ -8x² × 3

⇒ -24x²

Now middle term splitting :

★ Middle term : 5x

⇒ (8x - 3x)

★ Product of two parts :

⇒ 8x × -3x

⇒ -24x²

That means, we can now do middle term splitting using the two above parts :

⇒ 4√3x² + 5x - 2√3

⇒ 4√3x² + 8x - 3x - 2√3

Now taking common (4x) & (√3) common :

⇒ 4x(√3x + 2) - √3(√3x + 2)

[√3 × √3 = (√3)² = 3 ]

⇒ (4x - √3)(√3x + 2) [Required Solution]

Solution ❷ :

Given equation :

⇒ 7√2x² - 10x -4√2

Now dividing the middle term i.e. (-10x) into two such parts as told above :

★ Product of 1st & 3rd term :

⇒ 7√2x² × -4√2

⇒ -28x²(√2)²

⇒ -28x² × 2

⇒ -56x²

★ Middle term splitting :

⇒ -10x

⇒ (-14x + 4x)

★ Product of 2 parts :

⇒ -14x × 4x

⇒ -56x²

Now we can easily factorize the two parts using the two parts :

⇒ 7√2x² - 10x - 4√2

⇒ 7√2x² - 14x + 4x - 4√2

[Taking 7x & (4) common]

⇒ 7x(√2x - 2) + 4(x - √2)

[Taking √2 common ]

⇒ 7x × √2(x - √2) + 4(x - √2)

[√2 × √2 = (√2)² = 2 ]

⇒ 7√2x(x - √2) + 4(x - √2)

⇒ (7√2x + 4)(x - √2)

Hence, we get required solutions.