Quadratic formula — solve x² - 5x + 6 = 0 and discriminant analysis

Question

Solve $x^2 - 5x + 6 = 0$ using the quadratic formula. Find the discriminant and determine the nature of the roots. Verify your answer by factorisation.

(NCERT Class 10 — fundamental quadratic equation problem)

Solution — Step by Step

For $x^2 - 5x + 6 = 0$ :

a = 1, \quad b = -5, \quad c = 6

D = b^2 - 4ac = (-5)^2 - 4(1)(6) = 25 - 24 = 1

Since $D > 0$ , the equation has two distinct real roots.

Nature of roots based on discriminant:

$D > 0$ → two distinct real roots
$D = 0$ → two equal real roots (one repeated root)
$D < 0$ → no real roots (complex roots)

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

x = \frac{-(-5) \pm \sqrt{1}}{2(1)} = \frac{5 \pm 1}{2}

x = \frac{5 + 1}{2} = 3 \quad \text{or} \quad x = \frac{5 - 1}{2} = 2

The roots are $\mathbf{x = 3}$ and $\mathbf{x = 2}$ .

x^2 - 5x + 6 = (x - 2)(x - 3) = 0

We need two numbers that multiply to give $+6$ and add to give $-5$ . Those are $-2$ and $-3$ .

Setting each factor to zero: $x = 2$ or $x = 3$ . Matches our formula answer.

Why This Works

The quadratic formula is derived by completing the square on the general equation $ax^2 + bx + c = 0$ . It gives the exact roots of any quadratic equation, regardless of whether the equation can be factored neatly.

The discriminant $D = b^2 - 4ac$ appears under the square root. If $D$ is positive, the square root gives a real number, and the $\pm$ gives two different roots. If $D = 0$ , both roots collapse to the same value. If $D$ is negative, the square root of a negative number isn’t real — so no real roots exist.

Alternative Method — Factorisation (Splitting the Middle Term)

For equations with nice integer roots, factorisation is faster:

Find two numbers whose product = $a \times c = 1 \times 6 = 6$
And whose sum = $b = -5$
Those numbers: $-2$ and $-3$ (product = 6, sum = -5)
Split the middle term: $x^2 - 2x - 3x + 6 = x(x-2) - 3(x-2) = (x-2)(x-3)$

For CBSE boards: always show the discriminant analysis, even if you solve by factorisation. Questions often ask “find the nature of roots” separately — that requires the discriminant. Also, when $D$ is a perfect square and $a, b, c$ are rational, the roots are rational — this is a common follow-up question.

Common Mistake

Two frequent errors: (1) Students forget the negative sign in $-b$ . When $b = -5$ , we get $-b = -(-5) = +5$ , not $-5$ . Always be careful with double negatives. (2) Students divide only the numerator’s first term by $2a$ and forget the $\pm\sqrt{D}$ part. The entire expression $(-b \pm \sqrt{D})$ is divided by $2a$ — use brackets to avoid this error.

Question

Solution — Step by Step

Why This Works

Alternative Method — Factorisation (Splitting the Middle Term)

Common Mistake

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