Solve the quadratic equation (2x - 1)(3x + 7) = 0.

Step-by-step solution

1. 1

   Resist the urge to expand. The equation is already in the exact form you would work to reach if it were expanded. Multiplying it out to $6x^2+11x-7=0$ and then re-factoring is pure wasted effort.

2. 2

   Apply the zero product property. If a product of two real numbers is zero, at least one factor is zero:

   $AB=0\iff A=0\ \text{or}\ B=0$

   This property is what makes factoring a solution method at all, and it works only against zero — $(2x-1)(3x+7)=5$ would tell you nothing about the individual brackets.

3. 3

   Split into two linear equations.

   $2x-1=0\qquad\text{or}\qquad 3x+7=0$

4. 4

   Solve each one.

   $2x=1\ \Longrightarrow\ x=\frac{1}{2}$

   $3x=-7\ \Longrightarrow\ x=-\frac{7}{3}$

5. 5

   Substitute both roots back. At $x=\tfrac12$ the first bracket is $2(\tfrac12)-1=0$, so the product is $0$ $\checkmark$. At $x=-\tfrac73$ the second bracket is $3(-\tfrac73)+7=-7+7=0$, so the product is $0$ $\checkmark$. Both roots are exact.