Question

question
solve for x using the master product.
$2x^2 + 3x - 2 = 0$
master product:

Explanation:

Step1: Identify a, b, c

For quadratic \(2x^2 + 3x - 2 = 0\), \(a = 2\), \(b = 3\), \(c = -2\).

Step2: Find product \(a \times c\)

\(a \times c = 2 \times (-2) = -4\).

Step3: Find two numbers

Find two numbers that multiply to \(-4\) and add to \(3\). The numbers are \(4\) and \(-1\) (since \(4 \times (-1) = -4\) and \(4 + (-1) = 3\)).

Step4: Rewrite middle term

Rewrite \(3x\) as \(4x - x\): \(2x^2 + 4x - x - 2 = 0\).

Step5: Factor by grouping

Group terms: \((2x^2 + 4x) + (-x - 2) = 0\). Factor out GCF: \(2x(x + 2) - 1(x + 2) = 0\). Then \((2x - 1)(x + 2) = 0\).

Step6: Solve for x

Set each factor to zero: \(2x - 1 = 0\) or \(x + 2 = 0\). Solving, \(x = \frac{1}{2}\) or \(x = -2\).

Answer:

\(x = \frac{1}{2}\) or \(x = -2\)