Multiple choice

A quadratic equation, whose roots are $\alpha $ and $\beta$ can be written as $(x - \alpha ) (x - \beta) = 0 = x^2 - ( \alpha + \beta) x + \alpha \beta$ i.e. $ax^2 + bx + c \equiv a (x - \alpha) (x - \beta)$

  1. True

  2. False
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A Correct answer
Explanation

The identity (x - alpha)(x - beta) = x^2 - (alpha + beta)x + alpha*beta is the standard expansion for a monic quadratic equation with roots alpha and beta. Multiplying by a constant 'a' scales the equation to the general form ax^2 + bx + c.

AI explanation

According to the factor theorem in algebra, if alpha and beta are the roots of a quadratic polynomial, then (x - alpha) and (x - beta) are its factors. Expanding the product a(x - alpha)(x - beta) yields ax^2 - a(alpha + beta)x + a(alpha)(beta), which matches the standard form ax^2 + bx + c. Therefore, the given statement accurately describing the formation of a quadratic equation from its roots is true.

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