If the discriminant of a quadratic equation is \(D=(z-4)(z+6)\), which interval of \(z\) results in no real roots of the equation?
Answer and explanation
Correct answer: \(-6<z<4\)
A quadratic equation has no real roots when its discriminant satisfies \(D<0\). Thus, we solve \((z-4)(z+6)<0\). The product is negative between its zeros, \(-6\) and \(4\), so the correct interval is \(-6<z<4\). In option B, the product is positive, giving two real roots instead. Exam tip: for a product of two linear factors with positive leading coefficient, the sign is negative between the two zeros.
Frequently asked questions
What is the correct answer to this question?
\(-6<z<4\)
Why is this the correct answer?
A quadratic equation has no real roots when its discriminant satisfies \(D<0\). Thus, we solve \((z-4)(z+6)<0\). The product is negative between its zeros, \(-6\) and \(4\), so the correct interval is \(-6<z<4\). In option B, the product is positive, giving two real roots instead. Exam tip: for a product of two linear factors with positive leading coefficient, the sign is negative between the two zeros.
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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