If α and β are roots of x^2+4x-21=0, what is the value of |α-β|?
Answer and explanation
Correct answer: 10
The quadratic x^2+4x-21=0 factors as (x+7)(x-3)=0, because -7+3=-4 and (-7)(3)=-21. Thus its roots are 3 and -7. The absolute difference does not depend on which root is called α or β: |α-β|=|3-(-7)|=|10|=10. The same result follows from the discriminant. For a monic quadratic, the squared difference of its roots is Δ=b^2-4ac; here Δ=4^2-4(1)(-21)=16+84=100, so |α-β|=√100=10. Therefore option A is correct. The values 4, 7, and 21 are coefficients or related numbers, but none represents the separation between the two roots.
Frequently asked questions
What is the correct answer to this question?
10
Why is this the correct answer?
The quadratic x^2+4x-21=0 factors as (x+7)(x-3)=0, because -7+3=-4 and (-7)(3)=-21. Thus its roots are 3 and -7. The absolute difference does not depend on which root is called α or β: |α-β|=|3-(-7)|=|10|=10. The same result follows from the discriminant. For a monic quadratic, the squared difference of its roots is Δ=b^2-4ac; here Δ=4^2-4(1)(-21)=16+84=100, so |α-β|=√100=10. Therefore option A is correct. The values 4, 7, and 21 are coefficients or related numbers, but none represents the separation between the two roots.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Quadratic Equations. Topic: Roots of a Quadratic Equation.
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