Question

How do you factor `3x^{2}-48x+525`?

Answer 1

Factors of `3x^2-48x+525` are `3(x-8-isqrt111)(x-8+isqrt111)`

Explanation:

For factorizing quadratic polynomials, just find the value of discriminant. The discriminant for a quadratic polynomial `ax^2+bx+c` is `Delta=b^2-4ac` and then

the zeros of the polynomial are given by `(-b+-sqrtDelta)/(2a)` and factors of polynomial are `a(x-(-b-sqrtDelta)/(2a))(x-(-b+sqrtDelta)/(2a))`

In the quadratic polynomial `3x^2-48x+525`, we have `Delta=(-48)^2-4xx3xx525=2304-6300=-3996`

and roots are `(-(-48)+-sqrt(-3996))/6=(-(-48)+-6isqrt111)/6`,

where `i=sqrt(-1)`

and factors of `3x^2-48x+525` are

`3(x-(48-6isqrt111)/6)(x-(48+6isqrt111)/6)`

or `3(x-8-isqrt111)(x-8+isqrt111)`

Answer 2

`3x^2-48x+525 = 3(x-8-sqrt(111)i)(x-8+sqrt(111)i)`

Explanation:

One method involves completing the square.

We will also use the difference of squares identity:

`a^2-b^2 = (a-b)(a+b)`

with `a=(x-8)` and `b=sqrt(111)i` as follows:

`3x^2-48x+525 = 3(x^2-16x+175)`

`color(white)(3x^2-48x+525) = 3(x^2-16x+64+111)`

`color(white)(3x^2-48x+525) = 3((x-8)^2-(sqrt(111)i)^2)`

`color(white)(3x^2-48x+525) = 3((x-8)-sqrt(111)i)((x-8)+sqrt(111)i)`

`color(white)(3x^2-48x+525) = 3(x-8-sqrt(111)i)(x-8+sqrt(111)i)`