Question

write
${x}^{2}$
+8x+19 in the form of (x+a) whole square+b square what are the value of a and b ?​

Answer 1

$\bf{\large{\underline{\underline{Answer:-}}}}$

(x + a)² + b² = (x + 4)² + (√3)², value of a = 4 and value of b = √3

$\bf{\large{\underline{\underline{Explanation:-}}}}$

x² + 8x + 19

To get it in the form of (x + a)² + b²

First we need to write 8x as 2ab

So, 8x = 2(x)(4)

[Since 8 = 2 * 4]

We got 8x as 2(x)(4)

Now substitute 8x as 2(x)(4)

= x² + 2(x)(4) + 19

So, by (p + q)² = p² + 2pq + q²

We came to know that,

In the above expression p = x and q = 4

So now according to that identity we should get q² after 2(x)(4). we know the value of b i.e, 4. We can find the value of q² by squaring the value of q.

i.e, q² = (4)²

q² = 16

But according to the expression last term is 19 instead of 16 i.e, 4².

So, we need to split the the last term into two such that we can get 16.

i.e, 19 - 16 = 3

So, 19 = (16 + 3)

Substitute value of 19 as (16 + 3)

= x² + 2(x)(4) + (16 + 3)

= x² + 2(x)(4) + 16 + 3

Again 16 can be written as (4)²

= (x)² + 2(x)(4) + (4)² + 3

= [(x)² + 2(x)(4) + (4)²] + 3

We know that (p + q)² = p² + 2pq + q²

Here p = x and q = 4

By substituting the values in the identity we have,

= (x + 4)² + 3

Now here x = x, a = 4,

So, b² = 3

b = √3

x² + 8x + 19 = (x + 4)² + (√3)²

Now the expression is is the form of (x + a)² + b²

Therefore, (x + a)² + b² = (x + 4)² + (√3)², value of a = 4 and value of b = √3.

$\bf{\large{\underline{\underline{Identity\:Used:-}}}}$

(p + q)² = p² + 2pq + q²