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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