Question

Suppose one of the roots of the equation $a x^2 - b x + c = 0$ is $2 + \sqrt{3}$, where a, b and c are rational numbers and $a ≠ 0$. If $b = c^3$ then |a| equals

• A. 2
• B. 3
• C. 4
• D. 1

Answer 1

Answer: (A)

2

Answer 2

In a quadratic equation of the form $ax^2 + bx + c = 0$, the roots can be calculated using the formula: $x = \frac{(-b ± \sqrt{(b^2 - 4ac)}) }{ (2a)}.$

Since all the coefficients a, b, and c are rational, and one of the roots, 2 + √3, is irrational, it implies that the other root must also be irrational and be the conjugate of $2 + \sqrt{3}$, which is $2 - \sqrt{3}$

The sum of the roots can be found as: $\frac{(-b) }{ a} = (2 + \sqrt3) + (2 - \sqrt3) = 4.$

The product of the roots can be calculated as: $\frac{c }{ a} = (2 + \sqrt3)(2 - \sqrt3) = 4 - 3 = 1.$

However, it's important to note that the given quadratic equation is actually of the form $ax^2 - bx + c = 0.$

Hence, we can conclude that: $\frac{b }{ a} = \frac{4 c }{ a} = 1$

And it's also given that $b = c^3.$

Substituting $b = c^3$ into the equation $\frac{b }{ a} = 4$, we get: $\frac{(c^3) }{ a} = 4 c^3 = 4a$

Now, combining $\frac{c }{ a} = 1$ with $c^3 = 4a$, we have: $c^3 = 4 c = ∛4 c = ∓2$ (taking cube root on both sides)

So, we've found that $c = ∓2.$

Finally, calculating the absolute value of a, we get: |a| = 2.