NEED HELP PLEASE ANSWER ASAP! (41)

Question

asked May 24, 2023 78.3k views

2 Answers

Reynman · Answer 1 · 2023-05-25T16:09:12+0000

Answer:

41. (2) x² - 3x - 4 =0

Explanation:

Question 41

Standard form of polynomial

Sean Mickey · Answer 2 · 2023-05-28T08:33:57+0000

Answer:

(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
and sum to
, then rewrite
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
and sum to
.

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

Rewrite
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.