Question

Solve 9x2-9(a+b)x+(2a2+5ab+2b2)=0 by factorisation

Answer 1

$x=\frac{a+2 b}{3} \text { or } x=\frac{2 a+b}{3}$.    

To find:

Solve by Factorization: $9 \mathrm{x}^{2}-9(\mathrm{a}+\mathrm{b}) \mathrm{x}+\left(2 \mathrm{a}^{2}+5 \mathrm{ab}+2 \mathrm{b}^{2}\right)=0$

Solution:

$\begin{array}{l}{9 \mathrm{x}^{2}-9(\mathrm{a}+\mathrm{b}) \mathrm{x}+(2 \mathrm{a} 2+4 \mathrm{ab}+\mathrm{ab}+2 \mathrm{b} 2)=0} \\ \\ {9 \mathrm{x}^{2}-(9 \mathrm{a}+9 \mathrm{b}) \mathrm{x}+[2 \mathrm{a}(\mathrm{a}+2 \mathrm{b})+\mathrm{b}(\mathrm{a}+2 \mathrm{b})]=0}\end{array}$

$\begin{array}{l}{9 \mathrm{x}^{2}-(9 \mathrm{a}+9 \mathrm{b}) \mathrm{x}+[(\mathrm{a}+2 \mathrm{b})(2 \mathrm{a}+\mathrm{b})]=0} \\ \\ {9 \mathrm{x}^{2}-3[(\mathrm{a}+2 \mathrm{b})+(2 \mathrm{a}+\mathrm{b})] \mathrm{x}+\{(\mathrm{a}+2 \mathrm{b})(2 \mathrm{a}+\mathrm{b})\}=0}\end{array}$

$\begin{array}{l}{9 \mathrm{x}^{2}-3(\mathrm{a}+2 \mathrm{b}) \mathrm{x}-3(2 \mathrm{a}+\mathrm{b}) \mathrm{x}+\{(\mathrm{a}+2 \mathrm{b})(2 \mathrm{a}+\mathrm{b})\}=0} \\ \\ {3 \mathrm{x}[3 \mathrm{x}-(\mathrm{a}+2 \mathrm{b})]-(2 \mathrm{a}+\mathrm{b})[3 \mathrm{x}+(\mathrm{a}-2 \mathrm{b})]=0}\end{array}$

$[3 \mathrm{x}-(\mathrm{a}+2 \mathrm{b})][3 \mathrm{x}-(2 \mathrm{a}+\mathrm{b})]=0$

$[3 x-(a+2 b)]=0$ or we can have

$[3 x-(2 a+b)]=0$

$3 \mathrm{x}=(\mathrm{a}+2 \mathrm{b}) \text { or } 3 \mathrm{x}=(2 \mathrm{a}+\mathrm{b})$

Hence, $x=\frac{a+2 b}{3} \text { or } x=\frac{2 a+b}{3}$

Answer 2

Answer:

Step-by-step explanation:

Solution :-

Here, we have

9x² - 9(a + b)x + (2a² + 5ab + 2b²) = 0

Therefore,

Constant term = 2a² + 5ab + 2b²

= 2a² + 4ab + ab + 2b²

= 2a(a + 2b) + b(a + 2b)

= (a + 2b) (2a + b)

Coefficient of middle term = - 9(a + b) = - 3[(2a + b) + (a + 2b)

Then,

⇒ 9x² - 9(a + b)x + (2a² + 5ab + 2b²) = 0

⇒ 9x² - 3[(2a + b) + (a + 2b)]x + (2a + b) (a + 2b) = 0

⇒ 9x² - 3(2a + b)x - 3(a + 2b)x + (2a + b) (a + 2b) = 0

⇒ 3x[3x - (2a + b) - (a + 2b) [3x - (2a + b)] = 0

⇒ [3x - (2a + b)] [3x - (a + 2b) = 0

⇒ [3x - (2a +b)] = 0 or [3x - (a + 2b) = 0

⇒ x = 2a + b/3, a + 2b/3

Hence, x = 2a + b/3, a + 2b/3.