Quadratic Equations

A quadratic equation is an equation of the form ax² + bx + c = 0. 

Here, the only variable (we need to solve) is x. a, b and c are some real numbers which will be given so that we can solve for x. For example, 2x² + 3x + 4 = 0 is a quadratic equation with a = 2, b = 3 and c = 4. x² – 2x + 3 = 0 is another example with a = 1, b = –2 and c = 3. Of course, the variable needs not to be x, and a, b and c need not to be integers. For example, –4.9t² + 12.5t + 1.2 = 0 is also a quadratic equation with t as the variable. In fact, we can use this equation to solve the following problem: If you throw a ball, 1.2 meters above the ground, directly upward with a velocity of 12.5 meter per second, how long will it take to hit the ground?   

It should be mentioned that one side of the quadratic equation needs not to be 0. For example, x² + 2x = 3 and 2x² + 3 = 4x are also quadratic equations since they can be transformed into the form ax² + bx + c = 0: x² + 2x – 3 = 0 for the former and 2x² – 4x + 3 = 0 for the latter.

As we can see from the above, all these equation have one thing in common: the exponent of the variable is 2. Therefore, a quadratic equation is also known as a second degree equation or a degree-two equation. It should be also noted that a, the coefficient of x², can’t be 0. Since if a = 0, the equation ax² + bx + c = 0 becomes bx + c = 0, which is a first degree equation, not a second degree equation equation. However, either b or c can be 0. If b = 0, ax² + bx + c = 0 becomes ax² + c = 0 (click here to see how to solve this type of quadratic equations). If c = 0, ax² + bx + c = 0 becomes ax² + bx = 0 (click here to see how to solve this type of quadratic equations). Finally, click here to see how to solve ax² + bx + c = 0 in general.

Solving Quadratic Equations

Of the forms: 

ax² + c = 0 (where b = 0)

ax² + bx = 0 (where c = 0)

ax² + bx + c = 0 (where b and c ≠ 0)