Question

4p² + 7 = 12p

Answer 1

Given: 4p² + 7 = 12p
$\Rightarrow$4p² – 12p = –7
On dividing both sides by 4, we get:
p² – 3p = $\frac{-7}{4}$ … (1)
We make L.H.S. a perfect square by using

Third term = ${\left(\frac{1}{2}\times \mathrm{coefficient}\mathrm{of}\mathrm{p}\right)}^2={\left(\frac{1}{2}\times \left(-3\right)\right)}^2={\left(-\frac{3}{2}\right)}^2=\frac{9}{4}$

On adding 9/4 to both sides of equation (1), we get:

$$\begin{gathered} p^2-3p+\frac{9}{4}=\frac{-7}{4}+\frac{9}{4} \\ \Rightarrow p^2-2\times p\times \frac{3}{2}+{\left(\frac{3}{2}\right)}^2=\frac{-7+9}{4} \\ \Rightarrow {\left(p-\frac{3}{2}\right)}^2=\frac{2}{4} \end{gathered}$$

On taking square root of both sides, we get:

$$\begin{gathered} p-\frac{3}{2}=\pm \frac{\sqrt{2}}{2} \\ \Rightarrow p=\frac{3}{2}\pm \frac{\sqrt{2}}{2} \\ \Rightarrow p=\frac{3\pm \sqrt{2}}{2} \end{gathered}$$

Thus, $p=\frac{3\pm \sqrt{2}}{2}$.