How to solve quadratic equations — factoring, completing square, formula decision tree

Question

Given a quadratic equation $ax^2 + bx + c = 0$ , how do you decide whether to use factoring, completing the square, or the quadratic formula? Show each method with an example.

(CBSE Class 10 — quadratic equations carry 8-10 marks in boards)

Solution — Step by Step

Look for two numbers that multiply to give $ac$ and add to give $b$ .

Example: $x^2 + 5x + 6 = 0$

We need two numbers that multiply to 6 and add to 5 → 2 and 3.

x^2 + 2x + 3x + 6 = 0 \implies x(x+2) + 3(x+2) = 0 \implies (x+2)(x+3) = 0

\mathbf{x = -2 \text{ or } x = -3}

Use factoring when the numbers are “nice” — small integers that split easily.

Rearrange to make a perfect square on one side.

Example: $x^2 + 6x + 2 = 0$

x^2 + 6x = -2

(x^2 + 6x + 9) = -2 + 9

(x + 3)^2 = 7

x + 3 = \pm\sqrt{7}

\mathbf{x = -3 \pm \sqrt{7}}

The key step: add $\left(\frac{b}{2a}\right)^2$ to both sides to form the perfect square.

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

This works for any quadratic equation. The discriminant $D = b^2 - 4ac$ tells us about the roots:

$D > 0$ : two distinct real roots
$D = 0$ : two equal real roots (perfect square)
$D < 0$ : no real roots (complex roots — beyond Class 10)

flowchart TD
    A["Quadratic equation<br/>ax² + bx + c = 0"] --> B{Can you spot factors easily?}
    B -->|Yes| C["Factoring<br/>(fastest method)"]
    B -->|No| D{Is coefficient of x² = 1?}
    D -->|Yes| E["Completing the square<br/>(clean algebra)"]
    D -->|No or unsure| F["Quadratic formula<br/>(always works)"]
    C --> G["Check by substitution"]
    E --> G
    F --> G

Why This Works

All three methods solve the same equation — they just take different paths. Factoring relies on recognising patterns. Completing the square transforms the equation into a form where we can directly take square roots. The quadratic formula is actually derived by completing the square on the general equation $ax^2 + bx + c = 0$ — it is the universal version of Method 2.

The discriminant is powerful because it tells you about the nature of roots without solving the equation. If a board exam question asks “find the nature of roots,” just compute $D = b^2 - 4ac$ — no need to find the actual roots.

Alternative Method

For CBSE boards, the factoring method is expected for equations with integer roots. If you cannot factorise within 30 seconds, switch to the quadratic formula. Do not waste time trying to factor — the formula will always give the answer. Also, after finding roots, always verify by substituting back: if $x = r$ is a root, then $ar^2 + br + c$ should equal 0.

Common Mistake

Students forget the " $\pm$ " in the quadratic formula and write only one root. Every quadratic equation has two roots (they may be equal if $D = 0$ , but you must write both). Also, a common algebraic error: when $a \neq 1$ , students forget to divide the entire expression by $2a$ — they divide only the $-b$ part and leave $\sqrt{D}$ undivided. The entire numerator ( $-b \pm \sqrt{D}$ ) is divided by $2a$ .

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

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