Jack was out at a restaurant for dinner when the bill came his dinner came to $14 he wanted to leave a 31% tip

Find all the roots of the equation x^4-4x^3+x^2+12x-12=0

inysia [295]

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 -.
+12 / + 1
=> +1, +2, +3, +4, +6, +12 / +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 ≈ + 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!

3 0

3 years ago

Please help!!

podryga [215]

You can use the example and explanations below to make your portfolio.

1.One-step equations are equations that you can solve in just 1 step.
For example:

a. $x - 3 = 4$

To get x we use the ADDITIVE PROPERTY OF EQUALITY by adding the same quantity to both sides of the equation to cancel out figures and leave x. To cancel out 3 in the equation, we add 3 on both sides.

$x-3 + 3 = 4 +3$

-3 + 3 = 0
4+ 3 = 7

So your new equation would be:

$x + 0 = 7$

$x = 7$

Let's try another one step equation:

b. $\frac{x}{4} = 3$

In this example we will use another type of property that involves multiplication called MULTIPLICATIVE PROPERTY OF EQUALITY where we multiply both sides with the same quantity to cancel out the figure and leave x. In this case, the quantity on the left side of the equation that we need to cancel out is 4. So we multiply 4 on both sides of the equation.

$4 * \frac{x}{4} = 3 * 4$
$\frac{4x}{4} = 12$

We can cancel out 4 and that will leave you with just x.

$x = 12$

2. Equations with fractions:

To do equations with fractions you need to find the lowest common denominator, remove fractions by multiplying both sides with the LCD and solving for the unknown.

For example:

a. $\frac{3x}{2} = 6$

You can look at 6 on the right hand side as a fraction:
$\frac{3x}{2} = \frac{6}{1}$

Get the lowest common denominator of both denominators, which is 2 and 1 in this case. The LCD of 2 and 1 is 2. Now that you know it, you will multiply both sides with the LCD.

$2 * \frac{3x}{2} = \frac{6}{1} * 2$

Cancel the denominator on the LHS (Left-had side) and do the operation on the RHS (Right hand side). You will be left with

$3x = 12$

We can use another type of method to cancel out called transposing 3 to other side of the equation. When you do that you do the opposite operation. So if three is multiplied on one side, then when I transpose it it becomes division.

Your equation then will look like this:

$x = \frac{12}{3}$
$x = 4$

b. Let's try this on a more complicated equation:

$\frac{x + 3}{8} = \frac{2}{3}$

LCD of 8 and 3 is 24

$24 * \frac{x + 3}{8} = \frac{2}{3} * 24$

Simplify the expression on the LHS and RHS of the equation an you will be left with:
$3(x + 2) = 16$
$3x + 6 = 16$

Transpose 6 from the LHS to the RHS and its operation will become subtraction:
$3x= 16 - 6$
$3x= 10$

Divide both LHS and RHS by 3 to cancel out three in the LHS:
$\frac{3x}{3} = \frac{10}{3}$
$x = \frac{10}{3}$
or
$x = 3 \frac{1}{3}$

3. Distributive property:

If you noticed in the last example, we had a situation where one number is beside an equation enclosed in a parenthesis specifically:

3(x + 2)

If you see this, we use the distributive property first before moving on to solving the equation. Multiply the value that is outside the parenthesis with each number inside the parenthesis. Take note of the signs because you will consider it when multiplying. Let's use another example to do so:

3(x - 2) = 12

Distribute 3 to x and 2

3x - 6 = 12

Now you have a two-step equation:

Add 6 to both sides of the equation.

3x - 6 + 6 = 12 + 6
3x = 18

Divide both sides by 3:
3x/3 = 18/3
x = 6

4. Equations with decimals:

You can do equations with decimals as is but it is much easier if you clear the decimals first by making them into whole numbers. For example:

0.02x + 0.23 = 0.95

Notice that all decimals here are in the hundredths place, so to make them all whole numbers, you can multiply all decimals with 100 to make them whole. Take note that when you do this with one term, you have to do this for all terms to keep the statement true.

(100)0.02x + (100)0.23 = (100)0.95
2x + 23 = 95

Now that we have our new equation, we can solve for x much easier:
2x = 95 - 23
2x = 72
2x/2 = 72/2
x = 36

REAL WORLD EXAMPLE:
The shoe that you always wanted is on sale in the department store. It costs half the original price. The shoe now costs $25, what was the original price?
Equation:
$\frac{x}{2} = $25$
$x = $25 * 2$
$x = $50$

8 0

3 years ago

Read 2 more answers

Simplify (8x^2 − 1 + 2x^3) − (7x^3 − 3x^2 + 1).

DENIUS [597]

8x^2-1+2x^3-7x^3+3x^2-1

-5x^3+11x^2-2

8 0

3 years ago

Is 0.032 a terminating decimal

svp [43]

I don’t think it is, sorry I’m not 100% sure, good luck though

3 0