The value of k for which the quadratic equation, $kx^2+1=kx+3x-11x^2$ has real and equal roots are

Rearrange the equation to standard form by bringing all terms to one side, giving (k + 11)x squared minus (k + 3)x plus 1 equals 0. For the equation to have real and equal roots, the discriminant must equal zero, so (k + 3) squared minus 4(k + 11)(1) equals 0. Expanding and simplifying gives k squared plus 6k plus 9 minus 4k minus 44 equals 0, which results in k squared plus 2k minus 35 equals 0. Factoring this quadratic gives (k plus 7)(k minus 5) equals 0, so the values of k are 5 and negative 7.

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