Quadratic Equations
Logarithmic Equation — Sum of Real Solutions
nta_pyq_2026_jan
Grade 11
Question:
The sum of all the real solutions of the equation $\log_{(x+3)}(6x^2+28x+30)=5-2\log_{(6x+10)}(x^2+6x+9)$ is equal to
Step-by-Step Solution
Key Concept: Note: $6x^2+28x+30=2(3x+5)(x+3)$, $x^2+6x+9=(x+3)^2$, $6x+10=2(3x+5)$. Let $a=\log_{(x+3)}[2(3x+5)]$. LHS $=a+1$. $\log_{2(3x+5)}(x+3)^2=\tfrac{2}{a}$.
$x=1$ or $x=-1$. Sum $=0$.
Correct Answer: 3