COMPLETING THE SQUARE (PENYEMPURNAAN KUASA DUA)

IV. Uses
There are 2 uses for penyempurnaan kuasa dua:

Graphs

The equation is arranged in the form f(x) = a(x – p)2 + q

 

The turning point (titik pusingan) of the graph is at co-ordinates (p, q). So if you draw a graph for f(x) = 3(x – 5/6)2 + 8 11/12 the turning point is at (5/6, 8 11/12).

 

Also, a is positive, so the graph is u-shaped. If a is negative, the graph is n-shaped.

 

Warning: Look at the highlighted signs: f(x) = a(x p)2 + q. If the signs in your equation are different, then the co-ordinate will be negative.

 

f(x)

(x–4) 2 + 5

(x+4) 2 + 5

(x–4) 2 - 5

(x+4) 2 - 5

Coordinates

(4, 5)

(-4, 5)

(4, -5)

(-4, -5)

 

 

Solving quadratic equations

If you don’t want to divide EVERY number, you can bring the constant (pemalar) to the other side first, and divide the left side only.

 

2x2 – 5x + 2  = 0
2x2 – 5x  = -2
2[x25/2x]  = -2
2[x25/2x + (5/4)2 – (5/4)2]  = -2
2[(x – 5/4)225/16]  = -2
2(x – 5/4)225/8  = -2

 

You have now finished the ‘completing the square’ process, but you need to continue:

 

2(x – 5/4)225/8   = -2
2(x – 5/4)2   = -2 + 25/8
2(x – 5/4)2   = 9/8
(x – 5/4)2   = 9/16          <~( Divide both sides by 2)
x – 5/4   = √(9/16) or -√(9/16)
x – 5/4   = 3/or -3/4

 

The right side can be positive or negative, so you have two paths to take now.

x – 5/4  = 3/4
x  = 3/4 + 5/4
x  = 2
x – 5/4  = - 3/4
x  = - 3/4 + 5/4
x  = ½

 

So, for 2x2 – 5x + 2 = 0, the roots are 2 and 0.5

This technique is very tedious… Don’t use this unless you are instructed to.

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