Question

solve for x using the master product. $2x^2 + 9x + 9 = 0$ master product: try

Explanation:

Step1: Identify \(a\), \(b\), \(c\)

For quadratic \(ax^2 + bx + c = 0\), here \(a = 2\), \(b = 9\), \(c = 9\).

Step2: Calculate master product

Master product is \(a\times c\), so \(2\times9 = 18\).

Step3: Factor the quadratic

Find two numbers that multiply to 18 and add to 9. The numbers are 6 and 3.
Rewrite \(9x\) as \(6x + 3x\): \(2x^2 + 6x + 3x + 9 = 0\).
Factor by grouping: \(2x(x + 3) + 3(x + 3) = 0\), then \((2x + 3)(x + 3) = 0\).

Step4: Solve for \(x\)

Set each factor to zero:

• \(2x + 3 = 0\) gives \(x = -\frac{3}{2}\).
• \(x + 3 = 0\) gives \(x = -3\).

Answer:

The solutions are \(x = -3\) and \(x = -\frac{3}{2}\), and the master product is \(18\).