Please help me with steps

x = (3 − √5) / 2, (3 + √5) / 2, and 1

Step-by-step explanation:

Using rational root theorem, ±1 are possible roots.  By trial and error, we find that +1 is a root.  Therefore, x − 1 is a factor.  Knowing this, we can use grouping or long division.  Using grouping:

x⁴ − 5x³ + 8x² − 5x + 1 = 0

x⁴ − 5x³ + 4x² + 4x² − 5x + 1 = 0

x² (x² − 5x + 4) + 4x² − 5x + 1 = 0

x² (x − 4) (x − 1) + (4x − 1) (x − 1) = 0

(x² (x − 4) + 4x − 1) (x − 1) = 0

(x³ − 4x² + 4x − 1) (x − 1) = 0

Repeating the process, possible rational roots of x³ − 4x² + 4x − 1 are ±1.  We already know -1 isn't a root.  By trial and error, we find that +1 is again a root.  Using grouping to factor:

(x³ − 4x² + 3x + x − 1) (x − 1) = 0

(x (x² − 4x + 3) + x − 1) (x − 1) = 0

(x (x − 3) (x − 1) + x − 1) (x − 1) = 0

(x (x − 3) + 1) (x − 1) (x − 1) = 0

(x² − 3x + 1) (x − 1)² = 0

To find the roots of x² − 3x + 1, we use the quadratic formula.

x = [ -(-3) ± √((-3)² − 4(1)(1)) ] / 2(1)

x = [ 3 ± √(9 − 4) ] / 2

x = (3 ± √5) / 2

So the roots are x = (3 − √5) / 2, (3 + √5) / 2, and 1.

(2 x^3 + 7 x^2 + 5 x + 2)/(x (x + 2))

step-by-step explanation:

simplify the following:

x/x^2 + 3 x + 2 - (x (x + 1))/(x + 2)

combine powers. x/x^2 = x^(1 - 2):

x^(1 - 2) + 3 x + 2 - (x (x + 1))/(x + 2)

1 - 2 = -1:

x^(-1) + 3 x + 2 - (x (x + 1))/(x + 2)

put each term in 1/x + 3 x + 2 - (x (x + 1))/(x + 2) over the common denominator x (x + 2): 1/x + 3 x + 2 - (x (x + 1))/(x + 2) = (x + 2)/(x (x + 2)) + (3 x^2 (x + 2))/(x (x + 2)) + (2 x (x + 2))/(x (x + 2)) - (x^2 (x + 1))/(x (x + 2)):

(x + 2)/(x (x + 2)) + (3 x^2 (x + 2))/(x (x + 2)) + (2 x (x + 2))/(x (x + 2)) - (x^2 (x + 1))/(x (x + 2))

(x + 2)/(x (x + 2)) + (3 x^2 (x + 2))/(x (x + 2)) + (2 x (x + 2))/(x (x + 2)) - (x^2 (x + 1))/(x (x + 2)) = ((x + 2) + 3 x^2 (x + 2) + 2 x (x + 2) - x^2 (x + 1))/(x (x + 2)):

(2 + x + 3 x^2 (x + 2) + 2 x (x + 2) - x^2 (x + 1))/(x (x + 2))

3 x^2 (x + 2) = 3 x^3 + 6 x^2:

(2 + x + 3 x^3 + 6 x^2 + 2 x (x + 2) - x^2 (x + 1))/(x (x + 2))

2 x (x + 2) = 2 x^2 + 4 x:

(2 + x + 6 x^2 + 3 x^3 + 2 x^2 + 4 x - x^2 (x + 1))/(x (x + 2))

-x^2 (x + 1) = -x^2 - x^3:

(-x^2 - x^3 + 3 x^3 + 6 x^2 + 2 x^2 + 4 x + x + 2)/(x (x + 2))

grouping like terms, 3 x^3 - x^3 + 6 x^2 + 2 x^2 - x^2 + 4 x + x + 2 = (3 x^3 - x^3) + (6 x^2 + 2 x^2 - x^2) + (x + 4 x) + 2:

((3 x^3 - x^3) + (6 x^2 + 2 x^2 - x^2) + (x + 4 x) + 2)/(x (x + 2))

3 x^3 - x^3 = 2 x^3:

(2 x^3 + (6 x^2 + 2 x^2 - x^2) + (x + 4 x) + 2)/(x (x + 2))

6 x^2 + 2 x^2 - x^2 = 7 x^2:

(2 x^3 + 7 x^2 + (x + 4 x) + 2)/(x (x + 2))

x + 4 x = 5 x:

answer:   (2 x^3 + 7 x^2 + 5 x + 2)/(x (x + 2))

the correct option is 2.

step-by-step explanation:

according to the given information triangle abc is a right angle triangle with right angle at a. segment ab is 5 and segment ac is 7. point d is on segment bc and angles adb and adc are right angles.

it is given that triangles abd, cad, and cba are similar.

two triangles are similar if their corresponding sides are proportional.

in triangle cba, using pythagoras theorem

                      )

triangle abd and cba are similar,

                            (2)

triangle cad and cba are similar,

                      )

using (1), (2) and (3), we get

hence proved.

therefore option 2 is correct.