Find the value of $p$ for which the quadratic equation $x^2 + p(4x + p - 1) + 2 = 0$ has equal roots ?
- $\displaystyle -1, \frac{2}{3}$
- $3 , 5$
- $\displaystyle -1, \frac{4}{3}$
- $\displaystyle \frac{3}{4}, 2$
Equation: x^2 + 4px + p^2 - p + 2 = 0. For equal roots, discriminant D = (4p)^2 - 4(1)(p^2 - p + 2) = 0. 16p^2 - 4p^2 + 4p - 8 = 0. 12p^2 + 4p - 8 = 0. 3p^2 + p - 2 = 0. (3p-2)(p+1) = 0. Roots are p = 2/3, p = -1.
First, simplify the given equation x squared plus p times the quantity 4x plus p minus 1 plus 2 equals 0 to standard form, yielding x squared plus 4px plus p squared minus p plus 2 equals 0. For equal roots, the discriminant must be zero, so we set b squared minus 4ac equal to zero to get 4p squared minus 4 times 1 times the quantity p squared minus p plus 2 equals 0. Expanding and simplifying gives negative 4p plus 8 equals 0, which results in p equals negative 1 or p equals two thirds.