Solving Quadratic Equations by the Quadratic Formula
There are times when we are stuck solving a quadratic equation of the form ax2 + bx + c = 0 because the trinomial on the left side doesn’t factor nicely. It doesn’t mean that the quadratic equation has no solution. At this point, we need to call upon the straightforward approach of the quadratic formula to find the values of x.
In order use the quadratic formula, the quadratic equation that we are solving must be converted into the “standard form”, otherwise, all subsequent steps are wrong. It should look similar this…
In this convenient format, the numerical values of a, b, and c are easily identified! Upon knowing those values, we can now substitute them into the quadratic formula and then simplify to get the values of x.
Slow down if you need to. Be careful in every step while simplifying the expressions. This is where common mistakes usually happen because students tend to “relax” which results to errors that could have been prevented, such as in the addition, subtraction, multiplication and/or division of real numbers.
Example 1: Solve the quadratic equation below using the Quadratic Formula.
By inspection, it’s obvious that the quadratic equation is in the standard form since the right side is just zero while the rest of the terms stay on the left side. In other words, we have something like this…
This is great! What we need to do is simply identify the values of a, b, and c then substitute into the quadratic formula.
That’s it! Make it a habit to always check the solved values of x back into the original equation to verify.
Example 2: Solve the quadratic equation below using the Quadratic Formula.
This quadratic equation is absolutely not in the form that we want because the right side is NOT zero. I need to eliminate that 7 on the right side by subtracting both sides by 7. That takes care of our problem. After doing so, solve for x as usual.
The final answers are x1 = 1 and x2 = –2/3.
Example 3: Solve the quadratic equation below using the Quadratic Formula.
This quadratic equation looks like a “mess”. I have variable x‘s and constants in both sides of the equation. If we are faced with something like this, always stick to what we know. Yes, it’s all about the Standard Form. We have to force the right side be equal to zero. We can do just that in two steps.
I will first subtract both sides by 5x, and followed by the addition of 8.
Example 4: Solve the quadratic equation below using the Quadratic Formula.
Well, if you think that Example 3 is a “mess” then this must be even “messier”. However, if you really think about it, they are really very similar.
We need to perform some cleanup by converting this quadratic equation into standard form. Maybe you start realizing that this is not that bad as long as we know what to do, right?
Just to remind you, we want something like this…
Therefore we must do whatever it takes to make the right side of the equation equal zero. Since we have three terms on the right side, it follows that three steps are required to make it zero.
The solution below starts by adding both sides by 3x2, followed by subtraction of 3x, and finally addition of 5. Done!
After making the right side equal to zero, the values of a, b, and c are easy to identify. Plug those values into the quadratic formula , and simplify to get the final answers!
Example 5: Solve the quadratic equation below using the Quadratic Formula.
- Step 1: We need to rewrite the given quadratic equation equation in Standard Form,
Try doing it on your own first. Choose the correct answer from the choices below.
What did you get after working it out? Click below to see if you got the same answer.
Step 2: After getting the correct standard form in the previous step, it’s now time to plug the values of a, b and c into the quadratic formula to solve for x.
Try doing this again on your own. Choose the correct answer from the choices below.
Now let’s see if you got the same answer. Click the solution below.