# If a - 2b = 15 and ab = 11 then find the value of a^2 + 4b^2?

Jun 23, 2016

${a}^{2} + 4 {b}^{2} = 269$

#### Explanation:

${\left(a - 2 b\right)}^{2} = {a}^{2} - 4 a b + 4 {b}^{2} = {15}^{2}$

then

${a}^{2} + 4 {b}^{2} = {15}^{2} + 4 a b = {15}^{2} + 4 \times 11 = 269$

Jun 23, 2016

${a}^{2} + 4 {b}^{2} = 269.$

#### Explanation:

Method I

Given that, $a - 2 b = 15$
$\therefore {\left(a - 2 b\right)}^{2} = {15}^{2} = 225.$
$\therefore {a}^{2} - 2 \cdot a \cdot 2 b + 4 {b}^{2} = 225 ,$ i.e., ${a}^{2} - 4 a b + 4 {b}^{2} = 225.$

Letting, $a b = 11$ in this, we have, ${a}^{2} - 4 \left(11\right) + 4 {b}^{2} = 225.$

Hence, ${a}^{2} + 4 {b}^{2} = 225 + 44 = 269.$

Method II

Notice that ${\left(a + 2 b\right)}^{2} + {\left(a - 2 b\right)}^{2} = 2 \left({a}^{2} + 4 {b}^{2}\right) .$

Here, we replace ${\left(a + 2 b\right)}^{2}$ by ${\left(a - 2 b\right)}^{2} + 8 a b ,$ to get,

${\left(a - 2 b\right)}^{2} + 8 a b + {\left(a - 2 b\right)}^{2} = 2 \left({a}^{2} + 4 {b}^{2}\right) , .$ & now putting the given values,

$225 + 8 \cdot 11 + 225 = 2 \left({a}^{2} + 4 {b}^{2}\right) = 538 ,$ so, $\left({a}^{2} + 4 {b}^{2}\right) = \frac{538}{2} = 269 ,$ as before!