Substituting values of x
For many quadratic equations they can actually be solved by substituting values of x into the equation and then drawing the graph.
If you have time our advice is:
Always do a quick sketch
Example 1
Solve x2+3x-4=y
| x=2 | 22+3×2-4 | = | 4+6-4 | =6 |
| x=1 | 12+3×1-4 | = | 1+3-4 | =0 |
| x=0 | 02+3×0-4 | = | 0+0-4 | =-4 |
| x=-1 | (-1)2+3×(-1)-4 | = | 1-3-4 | =-6 |
| x=-2 | (-2)2+3×(-2)-4 | = | 4-6-4 | =-6 |
| x=-3 | (-3)2+3×(-3)-4 | = | 9-9-4 | =-4 |
| x=-4 | (-4)2+3×(-4)-4 | = | 16-12-4 | =0 |
| x=-5 | (-5)2+3×(-5)-4 | = | 25-15-4 | =6 |
If we plot these coordinates on a graph we get:
Now joining up the points we get:
Here we can see that the roots are +1 and -4
So why do we not use this way always?
Because you might get a beast like:
2x2-5x-6=y
Try plotting this one:
| x=4 | 2×42-5×4-6 | = | 32-20-6 | =6 |
| x=3 | 2×32-5×3-6 | = | 18-15-6 | =-3 |
| x=2 | 2×22-5×2-6 | = | 8-10-6 | =-8 |
| x=1 | 2×12-5×1-6 | = | 2-5-6 | =-9 |
| x=0 | 2×02-5×0-6 | = | 0-0-6 | =-6 |
| x=-1 | 2×(-1)2-5×(-1)-6 | = | 2+10-6 | =6 |
If we plot these coordinates on a graph we get:
Now joining up the points we get:
You can see that you can not find the roots graphically.
But you can find them using the quadratic formula.
Example 2
Solve 2x2-5x-6=y using the quadratic formula.
The quadratic formula is:
x=-b±√b 2-4ac2a
In this case a=2, b=-5 and c=-6
x=-(-5)±√(-5)2-4×2×(-6)2×2
x=5±√25+484
x=5±√734
x=5+√734 or x=5-√734
x=3.386 or x=-0.886
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