find the value of p so that the equation 3x2 - 5x = 2p = 0 has equal roots. also find the roots.

If the given quadratic equation is as:

 3x² - 5x + 2p = 0 then the solution is given as:

Solution:

We have, 

 3x² - 5x + 2p = 0

Now, for a given quadratic equation to have  real and equal roots its D = 0.

⇒ b² - 4ac = 0

⇒ (-5)² - 4 × 3 × 2p = 0

⇒ 25 - 24p = 0

⇒ -24p = -25

⇒ p = 25/24

$$\begin{gathered} \mathrm{Now}, \mathrm{p}\left(\mathrm{x}\right) = 3{\mathrm{x}}^2-5\mathrm{x}+\frac{25}{12} \\ \mathrm{Now}, \mathrm{zeroes} \mathrm{of} \mathrm{p}\left(\mathrm{x}\right) \mathrm{are} \mathrm{given} \mathrm{by} : \\ 3{\mathrm{x}}^2-5\mathrm{x}+\frac{25}{12} = 0 \\ \Rightarrow 36{\mathrm{x}}^2-60\mathrm{x}+25 = 0 \\ \Rightarrow {\left(6\mathrm{x}\right)}^2-2\left(6\mathrm{x}\right)\left(5\right)+{\left(5\right)}^2 = 0 \\ \Rightarrow {\left(6\mathrm{x}-5\right)}^2 = 0 \\ \Rightarrow 6\mathrm{x}-5 = 0 \mathrm{and} 6\mathrm{x}-5 = 0 \\ \Rightarrow \mathrm{x} = \frac{5}{6} \mathrm{and} \mathrm{x} = \frac{5}{6} \end{gathered}$$