The Quadratic Formula
In last week's blog post, I talked about two different ways of solving quadratic equations of the form ax²+bx+c=0. Some problems can be solved by factoring, and every problem can be solved by completing the square. However, as we saw in that post, completing the square can be pretty tough—it requires a lot of steps, and there are often several fractions involved.
For example, to complete the square on the equation 5x²+7x-8=0 takes about 13 steps! But since that particular expression isn't factorable, we seem to be stuck with that long, arduous process to solve the problem.
The good news is, there is another way! Instead of completing the square by hand, we can use a shortcut called the quadratic formula to do the work for us. All we have to do is plug the numbers from the equation into the formula to get the same answer we would have gotten by completing the square and solving the equation manually.
The formula says that if you have a quadratic equation that's in the form ax²+bx+c=0, then the answers for x will be $\frac{-b±\sqrt{b^2 - 4ac}}{2a}$.
For example, if we use the quadratic formula to solve 5x²+7x-8=0, then we get $\frac{-7±\sqrt{7^2 - 4(5)(-8)}}{2(5)}$. This simplifies to $\frac{-7±\sqrt{49 +160}}{10}$, which then becomes $\frac{-7±\sqrt{209}}{10}$. This is a much simpler and easier process than completing the square ourselves!
The quadratic formula can be found by completing the square on the generic equation ax²+bx+c=0, using the same steps we used to complete the square on any other equation. First, we factor out a from the first two terms; then we find h, which turns out to be $\frac{b}{a}÷2$, or $\frac{b}{2a}$. Then we add h² inside the parentheses and subtract h² times a outside the parentheses, and turn the parentheses into a perfect square. This gives us $a(x+\frac{b}{2a})^2 + c -a(\frac{b}{2a})^2=0$, which can simplify to $a(x+\frac{b}{2a})^2 + c -\frac{b^2}{4a})=0$. Now we solve for x using the reverse PEMDAS/"unwrapping" technique: first we add $-c+\frac{b^2}{4a})$ to both sides, and get a common denominator to turn that into $\frac{-4ac+b^2}{4a}$. Next, divide both sides by a, and square root both sides, remembering to include a ± to get both answers to the square root. Finally, we add $\frac{-b}{2a}$ to both sides, giving us $\frac{-b}{2a} ±\frac{\sqrt{-4ac+b^2}}{\sqrt{4a²}}$, which can simplify to $\frac{-b}{2a} ±\frac{\sqrt{-4ac+b^2}}{2a}$. Finally, since those have a common denominator, we can combine them into one fraction, giving us $\frac{-b±\sqrt{b^2 - 4ac}}{2a}$, which is the usual format for the quadratic formula.
Here's how the quadratic formula would look when applied to the examples from last week:
Example 1: x²+6 = -12x - 14
First, we add 12x+14 to both sides to get the equation into ax²+bx+c=0 form. Now we have x²+12x+20=0, so we can plug a=1, b=12, and c=20 into the formula, giving us $\frac{-12±\sqrt{12^2 - 4(1)(20)}}{2(1)}$. This simplifies down to $\frac{-12±\sqrt{64}}{2}$, which turns into -6±4. The positive case gives us x=-6+4, or -2, and the negative case gives us x=-6-4, which is -10.
Example 2: 4x²-40x-83=42
We start by getting zero on one side, by subtracting 42 from both sides. The equation becomes 4x²-40x-125=0, so we use the quadratic formula with a=4, b=-40, and c=-125. This gives us $\frac{40±\sqrt{(-40)^2 - 4(4)(-125)}}{2(4)}$, which simplifies to $\frac{40±60}{8}$. The positive version of that becomes 25/2, while the negative version becomes -5/2.
Example 3: 47x²-24x -8 = 12x²-6x
First we subtract 12x² and add 6x on both sides to get the equation into standard form. Now the equation is 35x² -18x -8=0, so we plug a=35, b=-18, and c=-8 into the quadratic formula, resulting in $\frac{18±\sqrt{(-18)^2 - 4(35)(-8)}}{2(35)}$. When we simplify that, it ends up becoming $\frac{18±38}{70}$, which gives us 4/5 for the positive case and -2/7 for the negative case.
Example 4: 6x²-5x = 2
We start by getting zero on one side, by subtracting 2 from both sides. The equation becomes 6x²-5x-2=0, so we use the quadratic formula with a=6, b=-5, and c=-2. This results in $\frac{5±\sqrt{(-5)^2 - 4(6)(-2)}}{2(6)}$, which simplifies down to $\frac{5±\sqrt{73}}{12}$. Since 73 isn't a perfect square, that can't be simplified any further, so the answers are $\frac{5+\sqrt{73}}{12}$ and $\frac{5-\sqrt{73}}{12}$.
Now you try! Test your knowledge by using the quadratic formula to solve each of the following equations:
6x²+14x-2
-7x²-18x+13
12x²-20x-8
