SAT High School

1. X⁵-4x⁴-2x³-2x³+4x²+x=0
2. X³-6x²+11x-6=0
3. X⁴+4x³-3x²-14x=8
4. X⁴-2x³-2x²=0
Find the roots for these problem show your work​

Answers

Answer 1

So the roots of the original equation are:

x = 0, x = 1 + √3, x = 1 - √3

Let's solve each of these equations and find their roots.

x⁵ - 4x⁴ - 2x³ - 2x³ + 4x² + x = 0:

To factorize this equation, we can factor out an "x" term:

x(x⁴ - 4x³ - 4x² + 4x + 1) = 0

Now, we have two factors:

x = 0

To find the roots of the second factor, x⁴ - 4x³ - 4x² + 4x + 1 = 0, we can use numerical methods or approximation techniques.

Unfortunately, this equation does not have any simple or rational roots. The approximate solutions for this equation are:

x ≈ -1.2385

x ≈ -0.4516

x ≈ 0.2188

x ≈ 3.4714

x³ - 6x² + 11x - 6 = 0:

This equation can be factored using synthetic division or by guessing and checking.

One possible root of this equation is x = 1.

By performing synthetic division, we can obtain the following factorization:

(x - 1)(x² - 5x + 6) = 0

Now, we have two factors:

x - 1 = 0

x = 1

x² - 5x + 6 = 0

To find the roots of the quadratic equation x² - 5x + 6 = 0, we can use the quadratic formula:

x = (-b ± √(b² - 4ac)) / (2a)

In this case, a = 1, b = -5, and c = 6.

Substituting these values into the quadratic formula, we get:

x = (5 ± √(25 - 24)) / 2

x = (5 ± √1) / 2

x = (5 ± 1) / 2

So the roots of the quadratic equation are:

x ≈ 2

x ≈ 3

Therefore, the roots of the original equation are:

x = 1, x ≈ 2, x ≈ 3

x⁴ + 4x³ - 3x² - 14x = 8:

To solve this equation, we need to move all the terms to one side to obtain a polynomial equation equal to zero:

x⁴ + 4x³ - 3x² - 14x - 8 = 0

Unfortunately, this equation does not have any simple or rational roots. We can use numerical methods or approximation techniques to find the roots.

Approximate solutions for this equation are:

x ≈ -2.5223

x ≈ -0.4328

x ≈ 1.6789

x ≈ 3.2760

x⁴ - 2x³ - 2x² = 0:

To solve this equation, we can factor out an "x²" term:

x²(x² - 2x - 2) = 0

Now, we have two factors:

x² = 0

x = 0

x² - 2x - 2 = 0

To find the roots of the quadratic equation x² - 2x - 2 = 0, we can again use the quadratic formula:

x = (-b ± √(b² - 4ac)) / (2a)

In this case, a = 1, b = -2, and c = -2. Substituting these values into the quadratic formula, we get:

x = (2 ± √(4 - 4(1)(-2))) / (2(1))

x = (2 ± √(4 + 8)) / 2

x = (2 ± √12) / 2

x = (2 ± 2√3) / 2

x = 1 ± √3

So the roots of the original equation are:

x = 0, x = 1 + √3, x = 1 - √3

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Related Questions

why are so many big cities located along the bank of a river

Answers

Answer: Rivers provide great and easy shipping routes and transportation of both goods and passengers.

Explanation:

Answer:I hope this help you out

Explanation:

One reason that many big cities are located along the bank of a river is because rivers have historically provided transportation as well as a source of water for drinking, agriculture, and industry. Additionally, rivers can be used for fishing, which has been an important food source for cities throughout history. Another reason is that the flat land around a river is often more accessible and easier to build on than hilly or mountainous terrain. Finally, rivers have traditionally been used for disposing of waste and sewage, which can be carried away downstream rather than accumulating in the city.

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