Which statement is always true about x^2-(a+3)x+3a=0?
Answer and explanation
Correct answer: 3 is always a root
To test whether 3 is always a root, substitute x=3 into the polynomial: 3^2-(a+3)(3)+3a=9-3a-9+3a=0 for every value of a. Therefore x=3 is always a root. The factor theorem then gives x^2-(a+3)x+3a=(x-3)(x-a), so the other root is a. This factorisation also shows why the remaining statements are false in general: a can itself be a root, the roots are equal only when a=3, and the equation has real roots for every real a because its roots are explicitly 3 and a. Thus option A is the only statement that holds without any restriction on a.
Frequently asked questions
What is the correct answer to this question?
3 is always a root
Why is this the correct answer?
To test whether 3 is always a root, substitute x=3 into the polynomial: 3^2-(a+3)(3)+3a=9-3a-9+3a=0 for every value of a. Therefore x=3 is always a root. The factor theorem then gives x^2-(a+3)x+3a=(x-3)(x-a), so the other root is a. This factorisation also shows why the remaining statements are false in general: a can itself be a root, the roots are equal only when a=3, and the equation has real roots for every real a because its roots are explicitly 3 and a. Thus option A is the only statement that holds without any restriction on a.
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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