1 is a root of the equation
\[x^2 + 4x -p = 0\].
The quadratic equation
\[kx^2 -2px +p=0\] has two equal roots.
Find the value of
\[k\].
Answer (1)
Here's how to solve the problem:Step 1: Find pSince 1 is a root of [tex]$\sf x^2 + 4x - p = 0$[/tex], we can substitute [tex]$\sf x=1$[/tex] into the equation:[tex]$\sf 1^2 + 4(1) - p = 0$[/tex][tex]$\sf 1 + 4 - p = 0$[/tex][tex]$\sf 5 - p = 0$[/tex][tex]$\sf p = 5$[/tex]Step 2: Use the discriminantThe quadratic equation [tex]$\sf kx^2 - 2px + p = 0$[/tex] has two equal roots. This means its discriminant is 0. The discriminant of a quadratic equation [tex]$\sf ax^2 + bx + c = 0$[/tex] is given by [tex]$\sf b^2 - 4ac$[/tex]. In our case, [tex]$\sf a = k$[/tex], [tex]$\sf b = -2p$[/tex], and [tex]$\sf c = p$[/tex]. Since [tex]$\sf p=5$[/tex], we have:[tex]$\sf (-2p)^2 - 4(k)(p) = 0$[/tex][tex]$\sf (-2(5))^2 - 4(k)(5) = 0$[/tex][tex]$\sf (-10)^2 - 20k = 0$[/tex][tex]$\sf 100 - 20k = 0$[/tex][tex]$\sf 20k = 100$[/tex][tex]$\sf k = \frac{100}{20}$[/tex][tex]$\sf k = 5$[/tex]Final answerTherefore, the value of k is 5.
