Solving Quadratic (2nd degree) Equations
There are 3 widely used methods for solving quadratic equations. A quadratic (or second-degree) equation is an equation in which the variable has an exponent of 2. The standard form of a quadratic equation is .
The three methods used to solve quadratic equations are: 1) factoring, 2) the square root property, and 3) the quadratic formula. Quadratic equations generally have 2 solutions.
1) Factoring is one method used
to solve second-degree and larger equations.
First the equation must be written in standard form. This means that the
polynomial must be in descending form and set equal to zero. Next, you
must factor the polynomial. (You may want to review
factoring.) Once the polynomial is factored, set any
factor which contains a variable equal to zero and solve (using isolation)
for the variable. Check your answer in the original equation. Every
quadratic equation has two solutions, although in the case of a perfect
square trinomial both of the solutions are the same.
2) The square root property involves taking the square roots of both sides of an equation. Before taking the square root of each side, you must isolate the term that contains the squared variable. Once this squared-variable term is fully isolated, you will take the square root of both sides and solve for the variable. We now introduce the possibility of two roots for every square root, one positive and one negative. Place a sign in front of the side containing the constant before you take the square root of that side.
the squared-variable term is isolated, so we
will take the square root of
notice the use of the sign, this will give us both a positive and a
again the squared-variable term is
isolated, so we will take the
this time p is not fully
isolated, also notice that 4 are
squared term is not isolated, add 1
to each side before
now take the square root of both sides
radical containing the constant cannot be
simplified, solve for the
notice the placement of the 1 before the
radical on the
In each of the first 3 examples involving the square root property, notice that there were no first-degree terms. These equations although they are quadratic in nature, have the form . To solve a quadratic equation that contains a first-degree term using the square root property would involve completing the square which is another "trick" that will be explained in another lesson.
method for solving quadratic equations described uses the quadratic formula.
first, the equation must be in standard form
identify a = 1, b = 8, c = 9
use the quadratic formula, substitute values
solve for r
and quadratic equations have 2 solutions
move terms to one side and set equation equal to zero
identify a = 5, b = 1, c = 1
cannot be simplified further
Remember ... quadratic equations generally have 2
The views and
opinions expressed in this page are strictly those of Mary Lou Baker. This page was
This page was edited on 28-Jun-2011