For which values of k will the roots of the equation x^2 = 2x(3k + 1) - 7(2k + 3) be equal?
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x^2 = 6kx + 2x - 14k - 21
x^2 - 6kx - 2x + 14k + 21 = 0
x^2 + (-6k-2)x + (14k+21) = 0
If the roots are equal, the discriminant must be zero:
(-6k-2)^2 - 4(1)(14k+21) = 0
36k^2 - 24k + 4 - 56k - 84 = 0
3k^2 - 80k - 80 = 0
k = (-(-80) +/- sqrt((-80)^2 - 4(3)(-80))) / (2*3)
k = (80 +/- sqrt(6400 + 960)) / 6
k = (80 +/- sqrt(7360)) / 6
k = (40 +/- sqrt(1840)) / 3
k =~ 27.6 or -0.965
x^2 - 6kx - 2x + 14k + 21 = 0
x^2 + (-6k-2)x + (14k+21) = 0
If the roots are equal, the discriminant must be zero:
(-6k-2)^2 - 4(1)(14k+21) = 0
36k^2 - 24k + 4 - 56k - 84 = 0
3k^2 - 80k - 80 = 0
k = (-(-80) +/- sqrt((-80)^2 - 4(3)(-80))) / (2*3)
k = (80 +/- sqrt(6400 + 960)) / 6
k = (80 +/- sqrt(7360)) / 6
k = (40 +/- sqrt(1840)) / 3
k =~ 27.6 or -0.965
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there are two ks in the equation. math is so dumb.