Quadratic Equations
Quadratic Equations
star_batch_jee_advanced_2025
Grade 11
Question:
If the equations $ax^3 + (-a + b)x^2 + (-b + c)x - c = 0$ and $2x^3 + x^2 + 2x - 5 = 0$ have a common root $(a \neq 0, a, b, c \in \mathbb{R})$ then $a + b + c$ is equal to:
Step-by-Step Solution
Key Concept: When two equations share a common root, analyze which roots are common and use the relationship between coefficients.
From $(x-1)(2x^2+3x+5)=0$ and $(x-1)(ux^2+bx+c)=0$, the common root is $x=1$. Since $2x^2+3x+5=0$ has imaginary roots, the relation between coefficients is $\frac{a}{2}=\frac{b}{3}=\frac{c}{5}$, which means $a+b+c=0$ or $\frac{a}{2}=\frac{b}{3}=\frac{c}{5}$.
Correct Answer: 1,2,4