Mathematics: Quadratics

Solving Quadratic Equations
Quadratic equations are a special set of trinomials of the form

that occur commonly in math and real-world applications. The roots of a quadratic equation are the solutions that satisfy the equation when
; in other words, where the graph touches the
-axis. There are several ways to determine these solutions including using the quadratic formula, factoring, completing the square, and graphing the function.

Quadratic Formula
The quadratic formula is used to solve quadratic equations when other methods are more difficult. To use the quadratic formula to solve a quadratic equation, begin by rewriting the equation in standard form
and <i>c</i> are coefficients. Once you have identified the values of the coefficients, substitute those values into the quadratic formula.
Evaluate the equation and simplify the expression. Again, check each root by substituting into the original equation. In the quadratic formula, the portion of the formula under the radical

is called the discriminant. If the discriminant is zero, there is only one root:
. If the discriminant is positive, there are two different real roots. If the discriminant is negative, there are no real roots, you will instead find complex roots. Often these solutions don't make sense in context and are ignored.

Factoring
To solve a quadratic equation by factoring, begin by rewriting the equation in standard form,
. Remember that the goal of factoring is to find numbers f and g such that
, in other words

and
.
This can be a really useful method when b and c are integers.
Determine the factors of c and look for pairs that could sum to b.

For example, consider finding the roots of
. The factors of -16 include,

and 4,

and
2,

and 8, -1 and 16, and 1 and -16. The factors that sum to 6 are

and

8. Write these factors as the product of two binomials,
. Finally, since these binomials multiply together to equal zero, set them each equal to zero and solve each for x. This results in
, which simplifies to

and
, which simplifies to
. Therefore, the roots of the equation are 2 and
.

Completing the Square
One way to find the roots of a quadratic equation is to find a way to manipulate it such that it follows the form of a perfect square (
) by adding and subtracting a constant. This process is called completing the square.

In other words, if you are given a quadratic that is not a perfect square,
, you can find a constant d that could be added in to make it a perfect square:




Once you have completed the square you can find the roots of the resulting equation:




It is worth noting that substituting the original expressions into this solution gives the same result as the quadratic formula where
:

Completing the square can be seen as arranging block representations of each of the terms to be as close to a square as possible and then filling in the gaps.

For example, consider the quadratic expression
:
 
 




Using Given Roots to Find Quadratic Equation
One way to find the roots of a quadratic equation is to factor the equation and use the zero product property, setting each factor of the equation equal to zero to find the corresponding root. We can use this technique in reverse to find an equation given its roots. Each root corresponds to a linear equation which in turn corresponds to a factor of the quadratic equation.
For example, we can find a quadratic equation whose roots are

and
.

The root

corresponds to the equation
, and the root

corresponds to the equation
.
These two equations correspond to the factors

and
, from which we can derive the equation
, or

Any integer multiple of this entire equation will also yield the same roots, as the integer will simply cancel out when the equation is factored. For example,

factors as
.

Solving a System of Equations Consisting of a Linear Equation and a Quadratic Equation Algebraically
Generally, the simplest way to solve a system of equations consisting of a linear equation and a quadratic equation algebraically is through the method of substitution. One possible strategy is to solve the linear equation for y and then substitute that expression into the quadratic equation. After expansion and combining like terms, this will result in a new quadratic equation for x which, like all quadratic equations, may have zero, one, or two solutions. Plugging each solution for x back into one of the original equations will then produce the corresponding value of y.

We can solve the linear equation for y to yield
.
Substituting this expression into the quadratic equation produces
. We can simplify this equation:

This quadratic equation can be factored as
. It therefore has two solutions:

and
. Plugging each of these back into the original linear equation yields

and
. Thus, this system of equations has two solutions,

and
.

It may help to check your work by putting each x and y value back into the original equations and verifying that they do provide a solution.
To solve a system of equations consisting of a linear equation and a quadratic equation graphically, plot both equations on the same graph. The linear equation will of course produce a straight line, while the quadratic equation will produce a parabola. These two graphs will intersect at zero, one, or two points; each point of intersection is a solution of the system.



The linear equation describes a line with a y-intercept of

and a slope of
.
To graph the quadratic equation, we can first find the vertex of the parabola: the x-coordinate of the vertex is
. Thus, the vertex lies at
. To get a feel for the rest of the parabola, we can plug in a few more values of x to find more points; by putting in

and

in the quadratic equation, we find that the points

and

lie on the parabola; by symmetry thus do

and
.

We can now plot both equations:


These two curves intersect at the points

and
, thus these are the solutions of the equation.

P1. Find the roots of
.
P2. Find a quadratic equation with roots

and
.

P1. First, substitute
0 in for y in the quadratic equation:

Next, try to factor the quadratic equation. Since
, list the factors of ac, or 8:

Look for the factors of ac that add up to b, or 8. Since none do, the equation cannot be factored with whole numbers. Substitute the values of a, b, and c into the quadratic formula,

:

Use the order of operations to simplify:



Reduce and simplify:






P2. The root

corresponds to the equation
, and the root

corresponds to the equation
. These two equations correspond to the factors

and
, from which we can derive the equation
, or