Quadratics

 

What is a Quadratic Expression?

 

A quadratic expression is any algebra expression where the variable is squared.  Often they are written in the format ax2 + bx + c where a, b and c are the coefficients.  Remember that x2 and x are not like terms, so an expression like 5x2 + 3x + 6 is in its simplest form.

 

Factoring Quadratic Expressions

 

Often a quadratic expression is the result of multiplying two algebraic expressions together, such as (x + 2)(x – 3).  The way you multiply these two expressions together is using the FOIL method.  FOIL stands for first, outer, inner and last.  It is a way of making sure you multiplied every part of the first expression by every part of the second.  The first amount in each set is x, so x • x = x2. The outer values are x and -3, so the second term is -3x.  The inner values are 2 and x, so the third term is 2x, and finally the last values are 2 and -3, which have a product of -6.  Now our quadratic expression is x2 - 3x + 2x – 6.  When we combine like terms we get x2 - x – 6.

 

Factoring an expression is doing the opposite.  For example, if we wanted to factor the above expression, x2 - x – 6, we know that the last two items in the two sets that were multiplied to get this expression have a product of -6.  We could make a list of all the things that have a product of -6:

 

1 and -6

-1 and 6

-2 and 3

-3 and 2

 

Of those, we know that we added the two coefficients of x to form the middle value, which is -1 (typically when you have -1x it is just written as –x).  Which of the above factors add up to -1?  -3 and 2 do, so they are the correct factors, and we can fill them into the sets (x – 3) and (x + 2)

 

Solving Quadratic Equations

 

When you solve for x in a quadratic equation, you will typically have two answers.  The reason for this has to do with the information that is lost when a number is squared.  Take this simple example:  x2 = 9.  We know that the square root of 9 is 3, so x could be 3, and, yes, 32 = 9.  But, -32 also is 9.  x2 = 9 has two solutions, 3 and -3.

 

Solving by Factoring

 

When you need to solve a quadratic equation, your first step should be to move all the terms to one side of the equals sign and set the equation equal to zero.  If you have the equation

 

x2 + 2x = 3

 

subtract 3 from each side to get

 

x2 + 2x - 3 = 0.

 

Now, you have a quadratic expression you can factor:  The factors of -3 are 3 and -1 or -3 and 1.  Since 3 and -1 add up to the middle number, 2, we know they are the correct factors, so right the equation factored:

 

(x – 1)(x + 3) = 0

 

Now consider what would make the factored expression equal 0.  1 would make it 0 because 1 – 1 = 0, and the other set would be multiplied by 0, and the expression would equal 0.  Likewise, -3 would make zero when added to 3.  Our possible answers are 1 and -3

 

 

Using the Quadratic Formula

 

In some cases, the equation may not be easily factorable.  This may be because there is a coefficient of x2 and factoring would take several steps, or it may be because the equation does not factor evenly, meaning one or both of your answers could be a fraction or decimal.  In these cases, we have a formula to help us.  On your formula sheet, you have two items that relate to quadratics.  The first is the standard form of a quadratic, where the coefficients of a simplified quadratic expression are shown as a, b and c:  ax2 + bx + c.  The a b and c in this expression tells us where to fill in the coefficients of the quadratic equation into the quadratic formula, which is the next on your formula sheet:

 

 

First, make sure your expression is set equal to zero.  Then, fill in the coefficient of x2 in place of a, the coefficient of x in place of b, and the value with no variable in place of c.  Make sure to include any negative signs.  When you get to the point in the equation when you have ±, split the expression into two parts, one with a + and one with a - .  You will then have your two answers.