S3T3 Solving Quadratic Equations by Factorisation/Formula

Recap

To solve for x, we ‘put’ x on the LHS (left-hand-side) and the number value to the RHS (Right-hand-side),

and we find the value of x.

Example 

            Solve for 7x + 3 = 17

                           7x = 17 – 3

                                = 14

                             x = 14/7 = 2

Solving Quadratic Equation

(1)  What is the meaning of  Solve x² + 3x + 2 = 0

=> to find the values (two or less) of x where it will make the equation = 0.

(2) Any number multiply by 0 is equal to 0.

=> For any (x + a)(x – b) = 0, as long as any one of (x + a) or (x – b) = 0, equation is solved because 

When x = -a

             (-a + a)( x- b) 

            = 0 x (x – b) = 0

When x = b

             (x+ a)( b - b) = 0 

              = (x + a) x 0 = 0

             = 0

(3) Solving the quadratic equation

            (x + P)(x – Q) = 0

        When (x + P) = 0

x + P = 0

x = -P

When (x – Q) = 0

    x – Q = 0

    x = Q

Values of x are 

         x = -P or x = Q

Quadratic equations are usually solved by putting the equation into the form

         (Jx + P)(Kx + Q) = 0 where J, K, P and Q are numerical values.

A quadratic equation can have 0, 1 or 2 values.

Recap : Factorisation  (x ² + bx + c)

Factorise x ² + 5x + 6

=> To find P and Q where   x ² + bx + c = x ² + 5x + 6 = (x + P)(x + Q)

     => b = 5 = P + Q, c = 6 = PQ 

Step 1 : Draw the cross X

<< Since is it x²  , fill in x² >>

                     x                [What is P?]

                            \ /

                            / \

                      x               [What is Q?]

Step 2 :  Find factors of c (P X Q) for b (P + Q)

When there is more than 1 set of factors,

Find the factor of c = 6 (P x Q)

6 = 1 x 6, 

          = 2 x 3

Using the X to cross-multiply, 

                     x          2                   x        1

                           \ /                             \ /

                           / \                             / \

                      x         3                    x        6

 x x 3 = 3 x, 2 x x = 2 x                x x 1 = 3 x, x x 6= 2 x

2 x + 3 x = 5 x                               x + 6 x = 7 x

P = 2 , Q = 3

Step 3 : Factorise

  =>              (  x        +2 )       

                            \ /

                            / \

                      ( x         +3)

 Thus,

                 x ² + 5x + 6 = (x + 2) (x + 3) 

Solving Quadratic Equation by Factorisation                                    

Example:

Solve y² + 3y + 2 = 0

            (y + 2)(y + 1) = 0                       By Factorisation     y  \  /        2

            (y + 2) = 0 or (y+ 1) = 0                                           y   /  \        1

            y = -2 or y = -1

Solving by Formula

For Equation:         ax2 + bx + c = 0

         formula x = -b +   √b2 – 4ac

                                          2a

Example:

Solve 2x²+ 6x + 1 = 0              (ax2 + bx + c = 0)

a = 2 , b = 6, c = 1

                      x = -6 +   √6²– 4 x 2 x 1

                                         2x2      

                        = -6 + √28

                                    4

                        = -6 + 2/4√7

                        = -6 + 7/2