How can I tell which method is best for solving a quadratic equation?
Quick answer
First write the equation in the form ax² + bx + c = 0. If it factors easily, use factoring. If completing the square keeps the arithmetic manageable, that can be a clear choice. If neither approach looks straightforward, use the quadratic formula: it works for any quadratic equation. Check your answers in the original equation.
The question, the details, the next step.
How can I tell which method is best for solving a quadratic equation? I know factoring, completing the square, and the quadratic formula, but I’m not sure how to choose.
Start by putting the equation in standard form, ax² + bx + c = 0, with a not equal to zero. Then look for the method that makes the least work:
- Try factoring if the terms suggest simple factors.
- Consider completing the square if rearranging the equation produces a manageable squared expression.
- If factoring would mean guessing, or completing the square looks cumbersome, use the quadratic formula. It works for any quadratic equation. [S1]
This is a practical choice, not a rule that you must always use one method for a particular equation. All three can lead to the solutions, but some make the arithmetic clearer than others.
What does “factors easily” look like? Could you show me a case where factoring is the sensible choice?
For a simple example, solve x² − 5x + 6 = 0. Look for two numbers whose product is 6 and whose sum is −5: −2 and −3. So the left side factors as (x − 2)(x − 3).
Set each factor equal to zero: x − 2 = 0 or x − 3 = 0. The solutions are x = 2 and x = 3. Substituting either value into the original equation gives zero.
Here, factoring is a quick choice because the pair of numbers is easy to spot. If you find yourself trying lots of pairs without progress, that’s a good reason to switch methods rather than keep guessing.
When is completing the square worth choosing instead? Can you work through an example without turning it into a general procedure?
It can be a useful choice when the x² and x terms readily form a square after rearranging. For example, consider x² + 4x − 1 = 0. Move the constant: x² + 4x = 1. Half of 4 is 2, and adding 2² to both sides gives x² + 4x + 4 = 5. The left side is (x + 2)², so (x + 2)² = 5.
Taking square roots gives x + 2 = ±√5, so x = −2 ± √5. This example keeps the arithmetic manageable. Completing the square involves more steps than the easy factoring example; if those steps become awkward, using the formula may be simpler.
What if I can’t see easy factors or a neat square? How do I use the formula as a fallback and check the result?
First identify a, b, and c in ax² + bx + c = 0, then use x = (−b ± √(b² − 4ac))/(2a). For 2x² + 2x − 3 = 0, a = 2, b = 2, and c = −3. Substitution gives x = (−2 ± √(4 + 24))/4 = (−2 ± √28)/4 = (−1 ± √7)/2.
The formula avoids needing to guess factors, though careful substitution and simplification still matter. To check, substitute each proposed value into the original equation and verify that the left side equals zero. The formula gives solutions; it does not by itself show that your copied coefficients or arithmetic were correct. For this equation, completing the square could also work, but the formula is a straightforward fallback when no method looks especially easy.
A practical checklist
Rewrite the equation as ax² + bx + c = 0 and identify a, b, and c.
Try factoring when a simple factor pair is apparent.
Consider completing the square if the rearranged expression stays manageable.
Use the quadratic formula when the other methods look awkward or uncertain.
Substitute each proposed solution into the original equation to check it.
A few more useful answers
Do I have to use the same method every time?
No. Choose based on the equation and the method you can carry out clearly. The quadratic formula works for any quadratic, while factoring can be quicker when the factors are easy to recognize. [S1]
Does completing the square work for every quadratic?
Completing the square can be used to solve quadratic equations, though it may take more steps than another method. For a dependable option that works for any quadratic, use the quadratic formula. [S1]
What if the equation is not written with zero on one side?
Rearrange it first so all terms are on one side and the other side is zero. Then identify the coefficients before choosing a method or substituting into the formula.
Sources
References used for this guide.
- [S1] OpenStax Elementary Algebra: Solve Quadratic Equations Using the Quadratic Formula openstax.org · Checked