Which quadratic polynomial has $6 + 7$ and $6 - 7$ as its zeroes?

Class 10PolynomialsAlgebra — Polynomials

Step-by-Step Solution

Sum = 12 and product =

(6)^{2} - (7)^{2} = 36 - 7 = 29

, so

p (x) = x^{2}

− (sum)

x

+ product =

x^{2} - 12 x + 29

. The constant 43 comes from computing 36 + 7 instead of applying

(a + b) (a - b) = a^{2} - b^{2}

to the product.

Answer: C)

x^{2} - 12 x + 29

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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