If $2x^2 + 12x + 18 = 0$, what is the value of $x$?

### Concept & Formula To solve a quadratic equation of the form $ax^2 + bx + c = 0$, we can simplify the equation by dividing by the greatest common divisor of the coefficients, and then factorize it or use the perfect square formula. ### Step-by-Step Solution Given equation: $$2x^2 + 12x + 18 = 0$$ Divide the entire equation by 2: $$x^2 + 6x + 9 = 0$$ Recognize the expression as a perfect square $(a + b)^2 = a^2 + 2ab + b^2$: $$(x + 3)^2 = 0$$ Taking the square root on both sides: $$x + 3 = 0 \implies x = -3$$ ### Exam Strategy & Shortcut Instead of solving completely, look at the signs. Since all signs in the equation are positive ($+12x, +18$), the value of $x$ must be negative to cancel them out. This immediately eliminates choices (c) and (d). Test $x = -3$: $$2(-3)^2 + 12(-3) + 18 = 18 - 36 + 18 = 0$$ This confirms the answer efficiently. ### Common Pitfall Students often confuse the sign during factorization, writing $(x - 3)^2 = 0$ instead of $(x + 3)^2 = 0$, leading to an incorrect answer of $3$. Always double-check by expanding your factored expression. ### Final Answer **Therefore, the correct answer is - 3.**