4.7.3 Completing the Square: An Example
The method of completing the square can be used to solve any quadratic equation.
With another tool in our arsenal, we are finally(!) ready to answer our original question, "When does the egg hit the ground?":
-4.9x^{2}+2.45x+1.7=0
To solve this equation we'll complete the square after getting rid of the leading coefficient; you may have noticed that all of the previous examples had a leading coefficient of 1. To achieve this, we can just divide both sides by -4.9. This leaves us with:
x^{2}-.5x-.3469=0
Now let's complete the square!
x^{2}-.5x=x^2-.5x + -
x^{2}-.5x+.0625-.0625-.3469=(x-.25)^2-.0625-.3469=0
Moving the -.0625 and the -.3469 to the right-hand side, we get that:
(x-.25)^2=.4094
We can then take the square root of both sides to see that
x-.25=\pm \sqrt{.4094}
Finally, we can add the .25 to both sides, to obtain the two solutions to the equation.
x is either or
But, our negative solution doesn't make sense since x is representing time.
So, we see that the egg hits the ground after:
.89 seconds.
We just spent a lot of time solving one particular quadratic equation. Along the way, however, we developed a tool that we can use to solve any quadratic equation. In other words, if instead of trying to solve -4.9x^{2}+2.45x+1.7=0, we had wanted to solve 2x^{2}+3x+4=0, we could have used the same exact process of completing the square to solve the equation. What this means for us is that we should be able to develop a process that works for any quadratic equation, ax^{2}+bx+c=0 where a, b, c are any constants (a≠0).
In the next section, we once again use the power of abstraction to develop a formula that will work for all quadratic equations. One of the most well-known tools in mathematics, it is known as the quadratic formula.
