Question

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

Accepted Answer

A. Since $a, b, c$ are rational numbers, irrational roots occur in conjugate pairs. Roots are $x_1 = 2 + \sqrt{3}$ and $x_2 = 2 - \sqrt{3}$. Sum of roots = $\frac{b}{a} = (2 + \sqrt{3}) + (2 - \sqrt{3}) = 4 \implies b = 4a$. Product of roots = $\frac{c}{a} = (2 + \sqrt{3})(2 - \sqrt{3}) = 4 - 3 = 1 \implies c = a$. Given $b = c^3$: $$4a = a^3$$ Since $a 
eq 0$, $a^2 = 4 \implies |a| = 2$.

Quadratic equation roots and coefficient ratio

Explanation

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