Quadratic Equations and Inequalities

Scan note

1) Solving quadratic equations means finding the roots of equation (the value of x)

2) Methods of solving quadratic equations:

(a) Factorisation.   (b) Completing the square.

(c) Quadratic formula.

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Before proceeding with this section we should note that the topic of solving quadratic equations will be covered in two sections. This is done for the benefit of those viewing the material on the web. This is a long topic and to keep page load times down to a minimum the material was split into two sections.

First Method :

(a) Factorisation

Before proceeding ,we should note the standard form of a quadratic equation.

       ax² + bx + c = 0 , a ≠ 0

The only requirement here is that we have an x² in this equation.We guarantee that this term will be present in the equation by requiring a ≠0. Note however,it is okay if b and / or c are 0.

•a / b / c can even be in a negative form ;

    

       -ax² - bx - c = 0 , a ≠ 0

Solving Steps

To solve a quadratic equation by factoring we first must move all the terms over to one side of the equation. Doing this serves two purposes. First, it puts the quadratics into a form that can be factored. Secondly, and probably more importantly, in order to use the zero factor property we MUST have a zero on one side of the equation. If we don’t have a zero on one side of the equation we won’t be able to use the zero factor properly.The value of x can be two different answers or can be two same answers.

As an example, the equation is:

    4x² + 9x + 5 = 5x + 3 

• In order to solve this equation ,we should put the quadratic into a form that can be factored.

•Regarding the general form of a quadratic equation,we should make the equation = 0 (by using algebra).

•If the equation can be simplified,we should simplify the equation.

From the equation above;

 4x² + 9x + 5 = 5x + 3

 4x² + 9x + 5 - 5x - 3 = 0

 4x² + 9x - 5x + 5 - 3 = 0

 4x² + 4x  + 2 = 0  «-- The equation can be simplified.

(4x² + 4x + 2 = 0) ÷ 2

 2x² + 2x + 1 = 0   «---The equation after simplified.

•Later, factorise the equation 

 [Alternative way : use calculator]

•By using calculator move the answer to make the equation = 0

 

Example ---» 2x² + 2x + 1 = 0

(from the | once you have inserted the      

equation. |value in calculator,

above).     |a=2 , b=2 ,c=1...  

                        

You'll get the values of x ,where is x = -1/2 ...  

Move the values after the = ,to make the equation = 0

--» x = -1/2 or x = -1/2

     2x = -1 or 2x = -1

     (2x + 1) = 0 or (2x + 1) = 0

~Therefore, the steps will be ;

   

   4x² + 9x + 5 = 5x + 3

   4x² + 9x + 5 - 5x - 3 = 0

   4x² + 9x - 5x + 5 - 3 = 0

  (4x² + 4x  + 2 = 0) ÷ 2

    2x² + 2x +1 = 0

  (2x+1)(2x+1) = 0

   2x+1 = 0

    2x = -1

      x = -1/2

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Second example :

    2x² - 8x +4 = 1 - x

• In order to solve this equation ,we should put the quadratic into a form that can be factored.

•Regarding the general form of a quadratic equation,we should make the equation = 0 (by using algebra).

•If the equation can be simplified,we should simplify the equation.

From the equation above;

   2x² - 8x + 4 = 1 - x

   2x² - 8x + 4 - 1 + x = 0

   2x² - 8x + x + 4 - 1 = 0

   2x² - 7x + 3 = 0 «---- This equation can't be                                          simplified.

•Later, factorise the equation 

 [Alternative way : use calculator]

•By using calculator move the answer to make the equation = 0

Example ---» 2x² - 7x + 3 = 0

(from the | once you have inserted the      

equation  |value in calculator,

above).    |a=2 , b=-7 ,c=3... 

You'll get the values of x ,which                        is x = 1/2 or x = 3...  Move the values after the = ,to make the equation = 0

--» x = 1/2 or x = 3

     2x = 1 or x = 3

     (2x - 1) = 0 or (x - 3) = 0

~Therefore, the steps will be ;

   2x² - 8x + 4 = 1 - x

   2x² - 8x + 4 - 1 + x

   2x² - 8x + x + 4 - 1 = 0

   2x² - 7x + 3 = 0

 (2x - 1)(x - 3) = 0

 2x - 1 = 0 or x - 3 = 0

  2x = 1 or x = 3

    x = 1/2 or x =3

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You can try these following questions.

[Exercise 1]

Move your mouse over the "Answer" to reveal the answer or click on the "Complete Solution" link to reveal all of the steps required for solving quadratics by factoring.

Solve: x2 – 13x = 30		Complete Solution
Solve: 20x2 – 130x = –200		Complete Solution
Solve: 4x(2x + 3) = 36		Complete Solution
Solve: 27x2 = 12		Complete Solution
Solve: (x + 3)2 – (2x – 1)2 = 0		Complete Solution
Solve: x(x + 4) = 32		Complete Solution
Solve: (2x – 5)(x + 1) = –12x		Complete Solution
Solve: (x – 7)2 + x2 = (x + 1)2		Complete Solution

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~•As mentioned at the start of this section we are going to break this topic up into two sections for the benefit of those viewing this on the web. The next two methods of solving quadratic equations, completing the square and quadratic formula, are given in the next section•~