Question

solve for x using the master product.
2x² - 3x - 9 = 0
master product: -18
answer attempt 1 out of 2
⊙ additional solution ⊙ no solution
x =

Explanation:

Step1: Identify a and c

For quadratic \(2x^2 - 3x - 9 = 0\), \(a = 2\), \(c=-9\). Master product: \(a\times c = 2\times(-9)=-18\).

Step2: Find factors of -18

Find two numbers that multiply to -18 and add to -3 (the coefficient of x). The numbers are -6 and 3 (\(-6\times3 = -18\), \(-6 + 3=-3\)).

Step3: Rewrite middle term

Rewrite the quadratic as \(2x^2 - 6x + 3x - 9 = 0\).

Step4: Factor by grouping

Group terms: \((2x^2 - 6x)+(3x - 9)=0\). Factor out common terms: \(2x(x - 3)+3(x - 3)=0\). Then \((2x + 3)(x - 3)=0\).

Step5: Solve for x

Set each factor to zero: \(2x+3 = 0\) or \(x - 3 = 0\). For \(2x+3 = 0\), \(2x=-3\), \(x = -\frac{3}{2}\). For \(x - 3 = 0\), \(x = 3\).

Answer:

\(x = 3\) or \(x = -\frac{3}{2}\)