Answer to Question #285504 in Algebra for bookaddict

Question

find the values of k for which the following equations have two equal roots.

(a)4x^2+4(k-2)x+k=0

(b)(k+2)x^2+4k=(4k+2x)

(c)x^2-2x+1=2k(k-2)

(d)(k+1)x^2+kx-2k=0

(e)4x^2-(k-2)x+9=0

Answer 1

(a)

$4x^2+4(k-2)x+k=0$

$D=(4(k-2))^2-4(4)(k)=0$

$k^2-4k+4-k=0$

$k^2-5k+4=0$

$(k-1)(k-4)=0$

$k\in \{1, 4\}$

(b)

$(k+2)x^2+4k=4k+2x$

$k\not=-2$

$(k+2)x^2-2x=0$

$x_1=0, x_2=\dfrac{2}{k+2}\not=0, k\in \R, k\not=-2$

There is no solution.

$k\in \empty$

(c)

$x^2-2x+1=2k(k-2)$

$(x-1)^2=2k(k-2)$

$2k(k-2)=0$

$k_1=0, k_2=2$

$k\in \{0,2\}$

(d)

$(k+1)x^2+kx-2k=0$

$k\not=-1$

$D=(k)^2-4(k+1)(-2k)=0$

$k^2+8k+8=0$

$k^2+8k+16=8$

$(k+4)^2=8$

$k_1=-2\sqrt{2}, k_2=2\sqrt{2}$

$k\in \{-2\sqrt{2},2\sqrt{2}\}$

(e)

$4x^2-(k-2)x+9=0$

$D=(-(k-2))^2-4(4)(9)=0$

$(k-2)^2=144$

$k_1=-10, k_2=14$

$k\in \{-10,14\}$