For the general quadratic equation ax^2+bx+c=0, which relation is correct when the roots are equal?
Answer and explanation
Correct answer: b^2=4ac
For ax^2+bx+c=0, where a is nonzero, the discriminant is Δ=b^2-4ac. The quadratic formula gives roots (-b±√Δ)/(2a). The roots are equal precisely when the plus and minus expressions coincide, which requires √Δ=0 and therefore Δ=0. Hence b^2-4ac=0, or b^2=4ac, so option A is correct. If b^2>4ac, the discriminant is positive and the roots are distinct real numbers. If b^2<4ac, the discriminant is negative and the roots are non-real conjugates. The relation a+b+c=0 instead only indicates that x=1 is a root; it does not generally imply equal roots.
Frequently asked questions
What is the correct answer to this question?
b^2=4ac
Why is this the correct answer?
For ax^2+bx+c=0, where a is nonzero, the discriminant is Δ=b^2-4ac. The quadratic formula gives roots (-b±√Δ)/(2a). The roots are equal precisely when the plus and minus expressions coincide, which requires √Δ=0 and therefore Δ=0. Hence b^2-4ac=0, or b^2=4ac, so option A is correct. If b^2>4ac, the discriminant is positive and the roots are distinct real numbers. If b^2<4ac, the discriminant is negative and the roots are non-real conjugates. The relation a+b+c=0 instead only indicates that x=1 is a root; it does not generally imply equal roots.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Quadratic Equations. Topic: Nature of Roots.
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