Definition: A quadratic equation is an equation that can be written as ax^2 + bx + c = 0 with a not equal to 0; it has at most two real solutions.
What makes an equation quadratic?
The highest power of the variable is 2. Examples are x2 = 49, 3x2 – x = 10, and (x + 2)(x – 5) = 8, which become standard form after expanding.
Which methods solve it?
- Square roots, when there is no x term.
- Factoring with the zero product property.
- Completing the square.
- The quadratic formula, which always works.
Example. Solve 5x2 – 80 = 0. Add 80 and divide by 5 to get x2 = 16. So x = 4 or x = -4. For x2 + 9x + 14 = 0, factoring gives (x + 2)(x + 7) = 0, so x = -2 or x = -7.
What goes wrong?
When taking a square root, learners keep only the positive answer. Another error is solving x2 = 5x by dividing by x, which loses the solution x = 0. Move terms and factor instead: x(x – 5) = 0 gives 0 and 5. Always check the answers in the original equation.
Example: x^2 – 12x + 35 = 0 has solutions x = 5 and x = 7.