Hi there!
The question here asks us to find all the roots, or solutions, of the quartic equation x⁴ - 4x³ + x² + 12x - 12 = 0. We can start off by using the Rational Root Theorem, which calls for the constant over the leading coefficient, which for us would be 12 over 1. Since we need to list all the factors of 12, they all become + or -.
<u>+</u>12 / <u>+</u> 1
=> <u>+</u>1, <u>+</u>2, <u>+</u>3, <u>+</u>4, <u>+</u>6, <u>+</u>12 / <u>+</u>1
Now, just simplify each one. We get the possible roots 1, -1, 2, -2, 3, -3, 4, -4, 6, -6, 12, -12. The next part is the most tedious. Plug in each possible factor for x and see if the resulting value is 0. We get that only root 2 makes the whole equation equal to zero. However, we are not done yet. The question asks for the ALL the roots of the equation, not just real, but also imaginary. Let's use synthetic division to find the other roots.
We can set the divisor to 2, as (x - 2) = 0 is equal to x = 2.
2| 1 -4 1 12 -12
2 -4 -6 12
1 -2 -3 6 0
We can now factor the whole original equation to (x - 2)(x³ - 2x² -3x + 6) = 0. Now, we have to factor the expression x³ - 2x² -3x + 6, and grouping seems like the most appropriate way.
x³ - 2x² -3x + 6 = 0
(x³ - 2x²) + (-3x + 6) = 0
x²(x - 2) - 3(x - 2) = 0
(x² - 3)(x - 2) = 0
Next, use the Zero Product Property to get the values of x.
x² - 3 = 0
x² = 3
x = √3
x ≈ <u>+</u> 1.73
x - 2 = 0
x = 2
Therefore, the roots of the equation x⁴ - 4x³ + x² + 12x - 12 = 0 are x = 2,x = 2 (repeated solution), x = 1.73, x = -1.73. Hope this helped and have a great day!
Both the numerator and the denominator are divisible by 2.
(4/2)/(10/2) so the simplest form would then be 2/5
Your answer is: 2/5
Step-by-step explanation:
y = -2x² + 4x + 1
Complete the square.
y = -2 (x² − 2x) + 1
y = -2 (x² − 2x + 1) + 1 − (-2)(1)
y = -2 (x − 1)² + 3
The vertex is (1, 3). The leading coefficient is negative, so the vertex is a maximum.
Graph: desmos.com/calculator/qsih3rd8ev
Answer:
B
Step-by-step explanation:
B. In the morning, Shawn added six hard candies to a jar which contained four hard candies.
So he had 10 candies.
By the end of the day he ate ten candies from this jar.
Now he had 10-10 = 0, zero candies.
Step-by-step explanation:
Suppose that the weight (in pounds) of an airplane is a linear function of the amount of fuel (in gallons) in its tank. When carrying 15 gallons of fuel, the airplane weighs 2187 pounds. When carrying 35 gallons of fuel, it weighs 2303 pounds. How much does the airplane weigh if it is carrying 50 gallons of fuel?
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You have two points relating fuel and weight.
(15,2187) and (35,2303)
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slope = (2303-2187)/(35-15) = 5.8
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intercept = ?
2187 = 5.8*15 + b
b = 2100
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Equation:
f(x) = 5.8x + 2100
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f(50) = 2390 lbs.
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