If alpha and beta are the zero of a quadratic polynomial p(x) find p(x) if alpha + beta=19 and alpha-beta=5

p(x) = x^12 -19x +84

Step-by-step explanation:

Adding the given equations yields ...

2·alpha = 19+5

alpha = 24/2 = 12

Then beta = 19-12 = 7

The factored form of p(x) is then ...

p(x) = (x -12)(x -7)

Multiplying this out gives ...

p(x) = x^2 -19x +84

Another way to get there is to realize that ...

p(x) = x^2 -(alpha+beta)x +(alpha·beta)

The constant term can be computed from the given sum and difference as ...

alpha·beta = ((alpha+beta)^2 -(alpha-beta)^2)/4 = (19^2 -5^2)/4 = 84

Then ...

p(x) = x^2 -19x +84

a complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is a solution of the equation x2 = −1. because no real number satisfies this equation, i is called an imaginary number. has no real solution, since the square of a real number cannot be negative.