Find the zeroes of polynomial p(x) = x² - 7x + 12 and verify sum/product relations

Question

Find the zeroes of the polynomial $p(x) = x^2 - 7x + 12$ and verify the relationship between the zeroes and the coefficients.

(NCERT Class 10, Chapter 2 — Polynomials)

Solution — Step by Step

We need two numbers that multiply to give $+12$ and add to give $-7$ .

Those numbers are $-3$ and $-4$ (since $(-3) \times (-4) = 12$ and $(-3) + (-4) = -7$ ).

x^2 - 7x + 12 = (x - 3)(x - 4)

Set each factor equal to zero:

$x - 3 = 0 \implies x = 3$

$x - 4 = 0 \implies x = 4$

The zeroes are $\alpha = 3$ and $\beta = 4$ .

For $ax^2 + bx + c$ , the sum of zeroes $= -b/a$ .

Here $a = 1$ , $b = -7$ , $c = 12$ .

Sum of zeroes $= \alpha + \beta = 3 + 4 = 7$

$-b/a = -(-7)/1 = 7$ ✓

Product of zeroes $= c/a$ .

$\alpha \times \beta = 3 \times 4 = 12$

$c/a = 12/1 = 12$ ✓

Both relationships are verified.

Why This Works

Every quadratic $ax^2 + bx + c$ can be written as $a(x - \alpha)(x - \beta)$ where $\alpha$ and $\beta$ are its zeroes. If you expand this:

a(x - \alpha)(x - \beta) = ax^2 - a(\alpha + \beta)x + a\alpha\beta

Comparing coefficients: $b = -a(\alpha + \beta)$ and $c = a\alpha\beta$ . That’s where the relations $\alpha + \beta = -b/a$ and $\alpha\beta = c/a$ come from. They’re not something to memorise blindly — they follow directly from factorisation.

Alternative Method — Using the Quadratic Formula

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{7 \pm \sqrt{49 - 48}}{2} = \frac{7 \pm 1}{2}

So $x = 4$ or $x = 3$ . Same answer.

For CBSE board exams, the factorisation method is faster and gets full marks. Use the quadratic formula only when the zeroes aren’t integers — for example, when the discriminant isn’t a perfect square.

Common Mistake

The most common error: students write the sum of zeroes as $b/a$ instead of $-b/a$ . That negative sign is critical. In this problem, $b = -7$ , so $-b/a = 7$ (not $-7$ ). Always write the formula first, then substitute — don’t try to do it in your head.

Question

Solution — Step by Step

Why This Works

Alternative Method — Using the Quadratic Formula

Common Mistake

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