SOLUTION: What is a step-by-step way to solve a Quadratic Equation?

Algebra ->  Quadratic Equations and Parabolas -> SOLUTION: What is a step-by-step way to solve a Quadratic Equation?      Log On


   

Question 234381: What is a step-by-step way to solve a Quadratic Equation?
Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!


There are three methods to solve a quadratic equation. For purposes of this discussion we will presume that the equation to be solved is in standard form, that is:



Factoring

If you can find numbers , , , and such that , , and , then you can write:



Then, using the Zero Product Rule, that is:

or

You can say:



Or



Completing The Square

Step 1: Put the equation into standard form, that is: .

Step 2: Multiply both sides of the equation by the reciprocal of the lead coefficient, that is: . becomes .

Step 2: Add the additive inverse of the constant term to both sides, that is: becomes .

Step 3: Divide the coefficient on the 1st degree term by 2 and then square the result, that is, calculate .

Step 4: Add the results of step 3 to both sides of the equation, that is: becomes

Step 5: Factor the perfect square on the left, that is: becomes .

Step 6: Take the square root of both sides. Remember to consider both the positive and negative roots, that is: becomes .

Step 7: Collect terms and simplify, that is: becomes .

The Quadratic Formula

The quadratic formula was derived by completing the square on the general quadratic -- just as was done in the discussion above on completing the square. Take the three coefficients, , , and and substitute them into the equation found in the last step of the completing the square discussion, namely:



Note that the character of the two roots is indicated by the Discriminant, which is the expression under the radical in the quadratic formula, that is:



Two real and unequal roots. If the discriminant is a perfect square, the roots are rational, otherwise they are irrational.

One real root with a multiplicity of two. That is to say that the trinomial is a perfect square and has two identical factors.

A conjugate pair of complex roots of the form where is the imaginary number defined by

John