Step 1: Equation Entry \[ x^2 + 5x + 6 = 0 \] Step 2: Quadratic Factorization Identify two numbers that satisfy the following conditions: - Their product equals the constant term (6). - Their sum equals the coefficient of \( x \) (5). Evaluate pairs of factors for 6: - \( 1 \times 6 = 6 \); \( 1 + 6 = 7 \) (Incorrect) - \( 2 \times 3 = 6 \); \( 2 + 3 = 5 \) (Correct) Therefore, the factorization is: \[ x^2 + 5x + 6 = (x + 2)(x + 3) \] Set each factor to zero: \[ x + 2 = 0 \implies x = -2 \] \[ x + 3 = 0 \implies x = -3 \] The roots are \( -2 \) and \( -3 \).
Step 3: Quadratic Formula Verification For an equation in the form \( ax^2 + bx + c = 0 \): \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] With \( a = 1 \), \( b = 5 \), and \( c = 6 \): \[ x = \frac{-5 \pm \sqrt{5^2 - 4 \cdot 1 \cdot 6}}{2 \cdot 1} \] \[ = \frac{-5 \pm \sqrt{25 - 24}}{2} = \frac{-5 \pm \sqrt{1}}{2} = \frac{-5 \pm 1}{2} \] Calculate the two possible values for \( x \): \[ x = \frac{-5 + 1}{2} = \frac{-4}{2} = -2 \] \[ x = \frac{-5 - 1}{2} = \frac{-6}{2} = -3 \] The roots are \( -2 \) and \( -3 \).
Step 4: Option Confirmation Option (1) \( -2 \) and \( -3 \) aligns with the calculated roots.