Question

solve for x using the master product.
$3x^2 - 7x + 4 = 0$
master product:

Explanation:

Step1: Identify a, b, c

For \(3x^2 - 7x + 4 = 0\), \(a = 3\), \(b = -7\), \(c = 4\). Master Product: \(a\times c = 3\times4 = 12\). Find two numbers that multiply to 12 and add to \(b=-7\). The numbers are -3 and -4 (since \(-3\times -4 = 12\) and \(-3 + (-4)=-7\)).

Step2: Rewrite middle term

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

Step3: Factor by grouping

Group terms: \((3x^2 - 3x)+(-4x + 4)=0\). Factor out common factors: \(3x(x - 1)-4(x - 1)=0\). Then \((3x - 4)(x - 1)=0\).

Step4: Solve for x

Set each factor to zero: \(3x - 4 = 0\) or \(x - 1 = 0\). For \(3x - 4 = 0\), \(3x = 4\), \(x=\frac{4}{3}\). For \(x - 1 = 0\), \(x = 1\).

Answer:

\(x = 1\) or \(x=\frac{4}{3}\)