PlosMathLearning

Quadratic Equations Quadratic Equations

Before doing this lesson, you should first familarize yourself with the lesson on quadratic expressions.

A quadratic equation is written in the form a2 + bx + c = 0.

You can solve a quadratic equation by using various methods. For now, we will be looking at two different methods:

1. Factorization

2. The Quadratic Formula


Method 1: Using the method of factorization

Example 1: Factorize a2 – 49 = 0.

Solution:

a2 – 49 = 0 can be written as

(a - 7)(a + 7) = 0

Either (a - 7 = 0 or a + 7 = 0

By solving these two equations, we get

a -7 = 0

a = 7

and a + 7 = 0

so, a = -7.

Therefore, our answer is a = 7 and a = -7.

You could also write your answer as a = ±7, which is read as "a is equal to plus or minus seven".


Example 2: Solve x2 + 7x + 12 = 0

Solution:

x2 + 7x + 12 = 0

x2 + 3x + 4x + 12 = 0

x(x + 3) + 4(x + 3) = 0

(x + 3)(x + 4) = 0

Either x + 3 = 0 or x + 4 = 0

By solving these two equations, we get

x + 3 = 0

x = -3

and x + 4 = 0

so, x = -4.

Therefore, our answer is a = -3 and a = -4.


Method 2: The Quadratic Formula

You must keep in mind that not all quadratic equations can be solved by factorization. The formula is a method that can be used to solved all quadratic equations.

The quadratic formula is derived from the general quadratic equation a2 + bx + c = 0.

The formula is given as x = -b ±b2 - 4ac
                                                        2a

Example 1: Solve x2 + 7x + 10 = 0.

There is no number written in front of the x2 term, but in that case it is helpful to think of the x2 term as 1x2 , so that we have:

        a = 1, b = 7 and c = 10

By substituting these values into the formula:

x = -b ±b2 - 4ac
               2a

We get,

x = -7 ±72 - 4(1)(10)
               2(1)

x = -7 ±49 - 40
               2

x = -7 ±9
           2

x = -7 ± 3
         2

Now, we can say,

x = -7 + 3
        2

x = -4
       2

  x = -2

      OR

x = -7 - 3
        2

x = -10
        2

  x = -5

So, your answer is x = -2 and x = -5.

Example 2: Solve 5x2 + 12x + 4 = 0.

In this example,

        a = 5, b = 12 and c = 4

By substituting these values into the formula:

x = -b ±b2 - 4ac
               2a

We get,

x = -12 ±122 - 4(5)(4)
               2(5)

x = -12 ±144 - 80
               10

x = -12 ±64
           10

x = -7 ± 8
         10

Now, we can say,

x = -12 + 8
        10

x = -4
       10

  x = -0.4

      OR

x = -12 - 8
        10

x = -20
        10

  x = -2

So, your answer is x = -0.4 and x = -2.




Copyright © 2012 Peter Smith

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