Solve the quadratic equation \(3x^2 - 5x - 2 = 0\) by factoring.
Answer
Finally, we solve for x by setting each factor equal to zero: \(3x + 1 = 0\) and \(x - 2 = 0\). Solving these equations gives us \(x = -1/3\) and \(x = 2\).
Step 1 ：First, we rewrite the equation in the form of \(ax^2 + bx + c = 0\), where a, b, and c are coefficients. In this case, the equation is already in this form, with a=3, b=-5, and c=-2.
Step 2 ：Next, we find two numbers that multiply to \(ac = (3)(-2) = -6\) and add to b=-5. The numbers -6 and 1 satisfy these conditions because \(-6 * 1 = -6\) and \(-6 + 1 = -5\).
Step 3 ：We rewrite the middle term of the equation as the sum of the terms -6x and 1x: \(3x^2 - 6x + x - 2 = 0\).
Step 4 ：We factor by grouping: \(3x(x - 2) + 1(x - 2) = 0\).
Step 5 ：We rewrite the equation as \((3x + 1)(x - 2) = 0\).
Step 6 ：Finally, we solve for x by setting each factor equal to zero: \(3x + 1 = 0\) and \(x - 2 = 0\). Solving these equations gives us \(x = -1/3\) and \(x = 2\).
