Question

NEED HELP PLEASE ANSWER ASAP! (41)

Answer 1

41. (2) x² - 3x - 4 =0

Explanation:

Question 41

Standard form of polynomial

• x² - (sum of roots) + product of roots = 0
• (2) x² - 3x - 4 =0

Answer 2

(2)
`x^2-3x-4=0`

Explanation:

Standard form of a quadratic equation:
`ax^2+bx+c=0`

When factoring a quadratic (finding the roots) we find two numbers that multiply to
`ac` and sum to
`b`, then rewrite
`b` as the sum of these two numbers.

So if the roots sum to 3 and multiply to -4, then the two numbers would be 4 and -1.

`\implies b=1+-4=-3`

`\implies ac=1 \cdot -4`

As there the leading coefficient is 1,
`c=-4`.

Therefore, the equation would be:
`x^2-3x-4=0`

Proof

Factor
`x^2-3x-4=0`

Find two numbers that multiply to
`ac` and sum to
`b`.

The two numbers that multiply to -4 and sum to -3 are: -4 and 1.

Rewrite
`b` as the sum of these two numbers:

`\implies x^2-4x+x-4=0`

Factorize the first two terms and the last two terms separately:

`\implies x(x-4)+1(x-4)=0`

Factor out the common term
`(x-4)`:

`\implies (x+1)(x-4)=0`

Therefore, the roots are:

`(x+1)=0 \implies x=-1`

`(x-4)=0 \implies x=4`

So the sum of the roots is: -1 + 4 = 3

And the product of the roots is: -1 × 4 = -4

Thereby proving that
`x^2-3x-4=0` has roots whose sum is 3 and whose product is -4.