Which condition is correct for real roots of ((k-2)x^2+2kx+(k+3)=0), if (k\neq2)?
Answer and explanation
Correct answer: (k\leq6) and (k\neq2)
For a quadratic equation \\(Ax^2+Bx+C=0\\), real roots exist when the discriminant \\(D=B^2-4AC\\) is non-negative. Here, \\(A=k-2\\), \\(B=2k\\), and \\(C=k+3\\). Therefore, \\(D=(2k)^2-4(k-2)(k+3)=4k^2-4(k^2+k-6)=4(6-k)\\). For real roots, \\(4(6-k)\\ge0\\), which gives \\(k\\le6\\).
The question separately states that \\(k\\ne2\\), because at \\(k=2\\) the coefficient of \\(x^2\\) becomes zero and the equation is no longer quadratic. Combining both conditions gives \\(k\\le6\\) and \\(k\\ne2\\). Hence option A is correct. The condition \\(D=0\\) is included because equal real roots are still real roots.
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What is the correct answer to this question?
(k\leq6) and (k\neq2)
Why is this the correct answer?
For a quadratic equation \\(Ax^2+Bx+C=0\\), real roots exist when the discriminant \\(D=B^2-4AC\\) is non-negative. Here, \\(A=k-2\\), \\(B=2k\\), and \\(C=k+3\\). Therefore, \\(D=(2k)^2-4(k-2)(k+3)=4k^2-4(k^2+k-6)=4(6-k)\\). For real roots, \\(4(6-k)\\ge0\\), which gives \\(k\\le6\\).
The question separately states that \\(k\\ne2\\), because at \\(k=2\\) the coefficient of \\(x^2\\) becomes zero and the equation is no longer quadratic. Combining both conditions gives \\(k\\le6\\) and \\(k\\ne2\\). Hence option A is correct. The condition \\(D=0\\) is included because equal real roots are still real 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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