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136 Mastering Technical Mathematics
Figure 10-13
At A, geometric
portrayal of quadratic
solutions
derived from completing
the square.
At B, the geometric
portrayal of the
solutions to the
first equation in
Fig. 10-11.
THE QUADRATIC FORMULA
Look at the significance of N 2 (x n) 2 . Here, x 2 is the big square. The final square
that represents N 2 is smaller, so x 2 is diminished by two quantities’ areas nx. The two
rectangles that represent nx overlap by another square that represents n 2 . So, as you
found in algebra, the geometric construction supports it:
N 2 x 2 2nx n 2
Figure 10-13B shows the geometric way of looking at the first equation in Fig. 10-11,
where the solutions are 3 or 3-1/3.
Look at the following two quadratic equations:
3x 2 4x 2 0
4x 2 3x 5 0
These equations are difficult to put into factored form. But there is a method known as
the quadratic formula that can be used to solve quadratics that are not “factor-friendly.”
Remember the general quadratic equation
ax 2 bx c 0
where a 0. The solution(s) to this equation can be found by using this formula:
x [b (b 2 4ac) 1/2 ] / 2a
The symbol is read “plus or minus” and is a way of compacting two mathematical
expressions into one. Written separately, the equations are
x [b (b 2 4ac) 1/2 ] / 2a
x [b (b 2 4ac) 1/2 ] / 2a