Question : Which of the following equations has equal roots?
Option 1: $3x^{2}-6x+2=0$
Option 2: $3x^{2}-6x+3=0$
Option 3: $x^{2}-8x+8=0$
Option 4: $4x^{2}-8x+2=0$
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Correct Answer: $3x^{2}-6x+3=0$
Solution : An equation has equal roots if its discriminant is zero. The discriminant of a quadratic equation $ax^{2}+bx+c=0$ is given by $b^{2}-4ac$. Option (i): $3x^{2}-6x+2=0$ So, $D=(-6)^{2}-4×3×2=36-24$ $\therefore D=12$ > 0 Option (ii): $3x^{2}-6x+3=0$ So, $D=(-6)^{2}-4×3×3=36-36$ $\therefore D=0$ Option (iii): $x^{2}-8x+8=0$ So, $D=(-8)^{2}-4×1×8=64-32$ $\therefore D=32$ > 0 Option (iv): $4x^{2}-8x+2=0$ So, $D=(-8)^{2}-4×4×2=64-32$ $\therefore D=32$ > 0 Hence, the correct answer is $3x^{2}-6x+3=0$.
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