Graphing is by far the easiest way to solve a quadratic equation -- but it does not always work and, when it works, the results can be inaccurate. If you have a computer with a math software package or a graphing calculator, though, it is a good place to start. Every place the graphed curve crosses the x axis is a root of the quadratic and, if the curve crosses the x axis at point p, x-p is a factor of the quadratic. There are two problems: the graphed curve may not cross the x axis, and if it does, it may be hard to read accurately.
Factoring is probably the most common way to solve quadratics. For example x^2+3x+2 = (x+1)(x+2). Once the factors are known, set each factor to zero and solve to get x=-1 and x=-2.
The problem is that not all quadratics are easy to factor -- especially quadratics with complex roots. These are the same ones whose graph did not cross the x axis. Factoring -- if it is easy -- is the first choice. Graphing first can help the factoring process.
Some quadratics are perfect squares -- the two factors are the same. The quadratic x^2+x+c is a perfect square if c=(b/2)^2. x^2-6x+8=0 is not a perfect square, but we can make it a perfect square by solving for x using a perfect square. In this example, 9 is the perfect square added to solve for x: x^2-6x+8=0; x^2-6x=-8; x^2-6x+9=-8+9; x^2-6x+9=1; (x-3)^2=1. So x-3=-1 and x-3=+1, or x=2 and x=4.
The quadratic formula always works, but it involves a lot more algebra than factoring quadratics that are easy to factor. The quadratic formula states that if ax^2+bx+c=0, the two roots are x=(-b-(b^2-4ac)*0.5)/2a and x=(-b+(b^2-4ac)*0.5)/2a. For example, if x^2-6X+8=0, the first root is x=(-(-6)-((-6)^2-4(1)(8))*0.5)/2(1)=2, and the other root is x=(-(-6)+((-6)^2-4(1)(8))*0.5)/2(1)=4.