Find the truth set of the equation x2 = 3(2x + 9)

Analyze the question: We need the values of x that satisfy x² = 3(2x + 9). This is a quadratic equation, so we rearrange it to the form ax² + bx + c = 0 and solve it by factorization.

Concept: To solve a quadratic equation by factorization, we find two numbers whose product is c (the constant term) and whose sum is b (the coefficient of x).

Solution:

Step 1: Expand the right-hand side. x² = 6x + 27

Step 2: Move all terms to one side. x² − 6x − 27 = 0

Step 3: Find two numbers whose product is −27 and whose sum is −6. Those numbers are −9 and +3, because (−9)(3) = −27 and −9 + 3 = −6.

Step 4: Factorize. x² − 9x + 3x − 27 = 0 x(x − 9) + 3(x − 9) = 0 (x − 9)(x + 3) = 0

Step 5: Solve for x. x − 9 = 0 gives x = 9 x + 3 = 0 gives x = −3

Check:

• x = 9: LHS = 81, RHS = 3(18 + 9) = 81 ✓
• x = −3: LHS = 9, RHS = 3(−6 + 9) = 9 ✓

The truth set is {x : x = −3, x = 9}, which is option D.

Tip: Always rearrange to "= 0" before factorizing, and verify by substituting your roots back into the original equation. This takes seconds and protects you from sign errors, which are the most common mistake in these questions. Also, the sign trick: if c is negative, your two numbers have opposite signs, and the larger one takes the sign of b.