For $x^{2} - 10 x + 21$ with zeroes α and β where α > β, what is the value of α − β?

Class 10PolynomialsAlgebra — Polynomials

Step-by-Step Solution

(α - β)^{2} = (α + β)^{2} - 4

αβ = 100 − 84 = 16, so α − β = 4 (the positive root, since α > β). Check: the zeroes are 7 and 3 ✓. Answering 16 stops at

(α - β)^{2}

and forgets to take the square root; 10 reports α + β instead.

Answer: A) 4

If α and β are zeros of

a x^{2} + b x + c

, then:

**α + β = −

b

/a** (sum of zeros).
**α · β =

c

/a** (product of zeros).

**Why**:

a x^{2} + b x + c = a (x

− α)

(x

− β) =

a

[

x^{2}

− (α+β)

x

+ αβ].
Matching coefficients with

x^{2} + (b / a) x + (c / a)

: coefficient of

x

→ −(α+β) =

b

/a → α+β = −

b

/a, and constant: αβ =

c

/a.

**Reverse use**: To form a quadratic with given zeros α, β, write

k

[

x^{2}

− (α+β)

x

+ αβ]
for any non-zero constant

k

(usually

k = 1

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