Three Different Methods for Solving Quadratic Equations

 

Method #1 – Factoring

If a quadratic equation has rational roots, then it can be easily factored.  This is often the easiest method to solve a quadratic equation but it only works if the roots are rational.  If the roots are irrational or complex you would have to use a different method.

 

	To factor with decomposition, we need to find two numbers that multiply to 6 (a x c) and add to 7 (the b value)… 1 and 6 would do.
	Now group the first two and the second two terms together and factor.
	Now for a second round of factoring
	Now to find the “zeros” or the roots of the equation.
	The solutions are x = -2 or -1/3

 

Method #2 – Completing the Square

This method is not used very often.  If the equation is hard enough not to factor then you will probably use the quadratic formula to solve it.  The method uses the exact same steps as changing from general to transformational form.  The only difference is that we have 0 instead of y and we take square roots at the end.

 

	Move the constant to the left side of the equation.
	Factor out the stretch (the “a” coefficient)
	Complete the square by adding half the “b” coefficient squared to both sides.
	Factor the perfect square
	Move the stretch to the other side (… divide both sides by the stretch of 3)
	Take the square root of both sides.  Remember the +/- bit for the square root...
	Solve for x!

 

 

 Method #3 – The Quadratic Formula

 

"In the land of Mordor, in the fires of Mount Doom, a formula was formed.  One formula to rule them all!”   This is the only equation you ever really need to solve quadratic equations.  Others might be quicker and easier (like factoring) but they won’t work for all equations.  The QF works every time.

 

	Make sure the equation is set equal to zero and find the coefficients. a = 3, b = 7 and c = 2.
	Plug the coefficients into the quadratic formula
	It's algebra from here to the finish line...    Solve it up!

 

Other Methods 

These are the “kickin’ it old skewl” methods for solving a quadratic equation.  If you have a TI-83 then you can graph the equation and use the Calculate Zeros command to find the solutions.  Of course the calculator will only give you approximate answers when there are irrational roots and an error if there are complex roots.  Even the TI-83 has its limitations…