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[x2 – 5/2x] | = -2 |
2[x2 – 5/2x + (5/4)2 – (5/4)2] | = -2 |
2[(x – 5/4)2 – 25/16] | = -2 |
2(x – 5/4)2 – 25/8 | = -2 |
You have now finished the ‘completing the square’ process, but you need to continue:
2(x – 5/4)2 – 25/8 | = -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 | = 3/4 or -3/4 |
The right side can be positive or negative, so you have two paths to take now.
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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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