SOLVE QUADRATIC EQUATIONS BY COMPLETING THE SQUARE

TIP: When doing these sort of problems, remember:

a) If you are certain that a quadratic equation cannot be factorised, then try completing
    the square.

b) Quadratics that can't be factorised can be solved either by completing the square or by     using the general quadratic formula.

c) If you end up trying to find the square root of a negative number, something has gone     wrong in your calculation - check your work again.

d) It's ok to leave your answer with the square root sign in (unless the questions asks for     answers to 2 d.p.). The square root sign is called a surd (which is not something you
    tread in curteousy of reckless dog owners!).

e) Here's how you complete the square for qudaratics beginning with x²:

e.g x² -6x +1 = 0

Step 1: Put the equation into the form p(x + q)² + r. To do this q needs to be half of -6.
           because (x - 3)² = x² - 6x + 9 which is close to what we want.

Step 2: To make the equation x² - 6x + 9 into x2 - 6x + 1, we need to subtract 8.
           so, we say that (x - 3)² - 8 is the same as x2 -6x + 1.

Step 3: Solve the equation (x - 3)² - 8 = 0.

(x - 3)²      = 8
(x - 3)        = Ö8
  x                = 3 ± Ö8

f) To solve quadratics where x² is prefixed by a number:

e.g. 2x² + 16x + 4 = 0     (2 is called a coefficient)

Step 1: Bring the 2 out of the equation by putting in brackets
           i.e. 2(x² + 8x + 2) = 0.            

Step 2: Divide both sides by 2 giving x² + 8x + 2 = 0.

Step 3: Complete the square as normal.

 

Solve the following quadratic equations by completing the square and give your answers in surd form: