Definition: The quadratic formula, x = (-b plus or minus the square root of (b^2 – 4ac)) / (2a), gives every solution of ax^2 + bx + c = 0.
When do you use it?
It works on every quadratic equation, including those that do not factor neatly. First write the equation in the form ax2 + bx + c = 0 so that a, b, and c are clear.
How do you apply it?
Substitute a, b, and c, keeping their signs. Evaluate the part under the root, called the discriminant, then compute both the plus and the minus case.
Example. Solve 3x2 – 5x – 8 = 0. Here a = 3, b = -5, c = -8. The discriminant is (-5)2 – 4(3)(-8) = 25 + 96 = 121, and its root is 11. So x = (5 ± 11) / 6. This gives x = 16/6 = 8/3 and x = -6/6 = -1.
What errors are common?
Forgetting that -b is +5 when b = -5. Dividing only the square root by 2a instead of the whole numerator. Also, if the equation is x2 = 7x – 2, move every term to one side first, or a, b, and c will be wrong. A negative discriminant means no real solutions.