Choosing the Best Method to Solve Quadratic Equations

Hello! Welcome to the fifth lesson in your Mathematics Mastery Sprint.

In the last lesson, we equipped ourselves with the three core algebraic methods for solving quadratic equations: factorization, the quadratic formula, and completing the square. We focused on mastering the mechanics of each technique.

Today, we're going to focus on strategy. You've mentioned that you sometimes find yourself choosing a difficult or inefficient method in exams, which can cost valuable time and lead to errors. This lesson is designed to fix that. Our goal is to build a clear, logical framework so you can instantly select the most efficient method for any given quadratic equation. This skill is crucial for improving your speed and confidence under exam pressure.

Learning Outcome: Select the most efficient method (factorization, formula, completing the square) to solve a given quadratic equation.

Your Strategic Roadmap for Solving Quadratics

Think of the three methods you've learned as different tools. A good mechanic doesn't use a sledgehammer to turn a small bolt. Likewise, you need to pick the right mathematical tool for the job.

Below is a flowchart that provides a powerful visual "road map" for this decision-making process. We will spend this lesson breaking down the logic behind it.

Flowchart for Solving Quadratic Equations — This flowchart outlines a systematic process for choosing the best method to solve a quadratic equation, starting from standard form, checking for simple cases, attempting factorization, and finally resorting to the quadratic formula or completing the square.

Let's dissect this process into a clear, hierarchical set of steps you can follow in an exam.

Step 1: Read the Question for Clues

Before you even look at the numbers, examine the wording of the question. The examiner often gives you explicit instructions or very strong hints.

The following resource, which is written specifically for your CIE IGCSE syllabus, details these clues perfectly.

Deciding the Quadratic Method

This revision note from Save My Exams is your go-to guide for this topic. It clearly lays out the 'when' and 'why' for each method based on typical exam question phrasing.

Read the three sections: 'When should I solve by factorisation?', 'When should I use the quadratic formula?', and 'When should I solve by completing the square?'. As you read, create a mental checklist based on the key phrases the guide points out.

Based on that reading, here is a summary of the key "command words" and clues:

Explicit Command: If the question says "Solve by factorisation" or "(a) Factorise... (b) Hence, solve...", you must use factorization. Similarly, if it says "(a) Write in the form $(x + p)^{2} + q$ ... (b) Hence, solve...", you must use your completed square from part (a).
Accuracy Clue: If the question asks for an answer "correct to 2 decimal places" or "3 significant figures," this is a huge signal that the equation will not factorize into simple numbers. You should immediately choose the quadratic formula.
No Clues: If the question just says "Solve..." with no other instructions, you have a choice to make. This leads us to the next step.

Step 2: Analyse the Equation's Structure

Once you've checked the instructions, look at the equation itself. Is it a full quadratic, or a simpler, two-term version?

Case 1: No $x$ term (i.e., $b = 0$ )
- Equation form: $a x^{2} - c = 0$
- Best Method: Don't use the full formula or try to factor! Just rearrange and take the square root. This is the fastest method.
- Example: To solve $2 x^{2} - 50 = 0$ :
  $2 x^{2} = 50$
  $x^{2} = 25$
  $x = \pm 25$
  $x = 5 or x = - 5$
Case 2: No constant term (i.e., $c = 0$ )
- Equation form: $a x^{2} + b x = 0$
- Best Method: Factor out the common factor of $x$ .
- Example: To solve $3 x^{2} - 12 x = 0$ :
  $x (3 x - 12) = 0$
  Then set each factor to zero:
  $x = 0 or 3 x - 12 = 0 ⟹ 3 x = 12 ⟹ x = 4$
- Warning: A common mistake here is to divide the whole equation by $x$ . If you do that, you will lose the $x = 0$ solution, and therefore lose marks.

Step 3: Attempt Factorization First

If you have a full trinomial ( $a x^{2} + b x + c = 0$ ) and the question doesn't give you any other clues, your default starting move should be to try factorization.

Why? It's almost always the quickest method if it works.
How long should you try? Don't spend more than 30-60 seconds. If you can't see the factors quickly, it's time to switch methods. This is a key time-management skill.

When is factorization a good bet?

When the coefficient $a = 1$ .
When the coefficients $a$ and $c$ are small integers with few factors.

Step 4: The Safety Net - The Quadratic Formula

The quadratic formula is your reliable backup. It works for every single quadratic equation.

Switch to the quadratic formula immediately if:

The question asks for an answer to a specific accuracy (as we saw in Step 1).
The coefficients $a, b,$ or $c$ are large or awkward numbers, making factorization tedious and prone to error.
Your 30-second attempt at factorization fails.

Don't see the formula as a "harder method". See it as the most reliable and systematic tool you have. Your focus should be on careful substitution, especially with negative signs.

This video gives a quick review of the decision criteria for all the main methods.

Solving Quadratic Equations (5 Methods)

This video from Mario's Math Tutoring provides a concise summary of when to use each of the main solving methods. Watching this will help solidify the decision-making process in your mind.

Watch the video from the beginning (0:00 - 08:20). For each of the five methods discussed (graphing is less common in exams unless the graph is given), pay attention to the presenter's initial explanation of when you would choose that method.

Let's Practice the Strategy

Now, let's put this into practice. For each equation below, decide which method you would use and, most importantly, why. Don't solve them yet—the goal is to practice the decision-making process.

Problem 1

Question: Solve $x^{2} - 5 x - 14 = 0$ .

Show your reasoning

Method: Factorization.
Reasoning: There are no instructions about accuracy. The equation is a full trinomial with $a = 1$ . I should look for two numbers that multiply to -14 and add to -5. These are easy to spot as -7 and +2. This will be very fast.
- Solution would be $(x - 7) (x + 2) = 0 ⟹ x = 7 or x = - 2$ .

Problem 2

Question: Find the solutions to $3 x^{2} + 7 x = 9$ , giving your answers correct to 3 significant figures.

Show your reasoning

Method: Quadratic Formula.
Reasoning: The instruction "correct to 3 significant figures" is a direct command to use a method that handles non-integer answers. Factorization will not work. First, I would rearrange to $3 x^{2} + 7 x - 9 = 0$ and then substitute $a = 3, b = 7, c = - 9$ into the formula.

Problem 3

Question:
(a) Write $x^{2} + 8 x - 5$ in the form $(x + p)^{2} + q$ .
(b) Hence, or otherwise, solve the equation $x^{2} + 8 x - 5 = 0$ .

Show your reasoning

Method: Completing the Square.
Reasoning: The word "Hence" in part (b) strongly implies I should use the result from part (a). Part (a) explicitly asks me to complete the square. I would set my completed square form equal to zero and solve from there. This is much faster than starting again with the formula.

Problem 4

Question: Solve $6 x^{2} - 13 x + 6 = 0$ .

Show your reasoning

Method: A strategic choice between Factorization and the Quadratic Formula.
Reasoning: There are no accuracy instructions, so factorization is possible. However, with $a = 6$ , finding the factors might take a moment ( $6 \times 6 = 36$ , need two numbers that multiply to 36 and add to -13... which are -4 and -9).
- Path A (Confident Factorizer): I spot the factors quickly and proceed. $(2 x - 3) (3 x - 2) = 0 ⟹ x = 3/2 or x = 2/3$ .
- Path B (Safe & Systematic): The factors aren't immediately obvious, and I don't want to waste time or make a mistake. I will switch to the quadratic formula with $a = 6, b = - 13, c = 6$ . It might take a few more seconds of calculation, but it is a guaranteed path to the correct answer.
Conclusion: Both are valid, but recognizing when to abandon factorization for the safety of the formula is a top-level exam skill.

Conclusion

You now have a strategic framework for tackling any quadratic equation. By internalizing this process, you will stop choosing inefficient methods and start saving time and securing marks reliably.

Key Takeaways: Your Decision Hierarchy

Check the question's wording: Look for commands ("factorise", "complete the square") or clues ("2 decimal places"). Let the examiner guide you.
Check the equation's structure: If $b = 0$ , rearrange and square root. If $c = 0$ , factor out $x$ .
Attempt to factorize first: For full trinomials with no other clues, give factorization a quick try (30-60 seconds).
Use the formula as your default safety net: If factorization fails, is too complex, or if accuracy is required, switch to the quadratic formula without hesitation. It always works.

Preview of the Next Lesson:
In the next lesson, we will focus on rearranging complex formulas to change the subject, particularly when the subject appears twice. For example, making $x$ the subject of $y = x^{2} + 6 x$ . You'll see that the skill of completing the square, while not always the fastest for solving, is the essential tool for this next important algebraic manipulation.

Recommended resources

Your Strategic Roadmap for Solving Quadratics

Step 1: Read the Question for Clues

Step 2: Analyse the Equation's Structure

Step 3: Attempt Factorization First

Step 4: The Safety Net - The Quadratic Formula

Let's Practice the Strategy

Conclusion