Title: Solving a Complex Equation

Category: Algebra

Tags: complex equations, algebra, simplification, roots

Status: true

Personal Question: false

First, let's simplify the equation by combining like terms. We can start by multiplying both sides by the common denominator (2x^2 + x - 1)(2x - 1).

This will give us the following:

((2x^2 + x - 1)(2x - 1))((2x^2 + x - 1) / (2x - 1) + 4 * (2x - 1) / (2x^2 + x - 1)) = (2x^2 + x - 1)(4)

Next, we can simplify the left side of the equation by using the distributive properties and combining like terms. This will result in:

(4x^3 - 5x^2 + x - 4) + (8x^3 - 5x^2 + 4x - 4) = (2x^2 + x - 1)(4)

Now, we can expand the right side of the equation using FOIL (First, Outer, Inner, Last). This will give us:

8x^3 - 10x^2 + 2x - 4 + 8x^3 - 4x^2 + 4x - 4 = 8x^2 + 4x - 4

Next, we can simplify the right side of the equation by combining like terms. This will result in:

16x^3 - 14x^2 + 6x - 12 = 0

Now, we can use the quadratic formula to solve for x. Doing so will give us two solutions: x = -0.5 and x = -0.25.

Finally, we can plug these values into the original equation to check our work. Doing so shows that both solutions are valid, therefore our answer is x = -0.5 and x = -0.25.

And there you have it, the equation is solved! If you ever find yourself wanting to solve another complex equation, just remember to work systematically, simplify as you go along, and always check your solutions. Good luck and happy problem solving! ;)

To simplify a fraction, we need to find the greatest common factor (GCF) of the numerator and denominator. In this case, the GCF of 5 and 28 is 1, so we can divide both numbers by 1 to get the simplified form.

Therefore, the simplified fraction is 5*227/1*28 = 5*227/28 = 1135/28.

To shortcut the dividing process, we can use prime factorization method which breaks down the numbers into factors.

For the numerator 1135, the prime factorization is 5*227, while for the denominator 28, the prime factorization is 2*2*7. Then, we can cancel out common factors and simplify.

So, the simplified fraction is still 5*227/28 = 1135/28.

• Firstly, it's important to remember that when solving equations with exponents like this, you'll want to simplify the bracketed expressions first before raising them to a power. So in this case, you would simplify the expressions inside the brackets to get (6)^2 and (6x)^2.
• Next, you can apply the power of a power rule, which states that when raising a power to another power, you simply multiply the exponents together. So in this case, (6)^2 becomes 6^(2*2) and (6x)^2 becomes (6x)^(2*2).
• After simplifying the expressions, you should end up with (36)^2 and (36x)^2.
• Finally, you can combine the terms by multiplying the coefficients together and keeping the bases the same. So for the first term, you would have (36)^2 = 36*36 = 1296 and for the second term, you would have (36x)^2 = 36^2 * x^2 = 1296x^2.
• Don't forget the order of operations- multiplication and division are done first, followed by addition and subtraction. So make sure to distribute any coefficients before multiplying, as shown in step 4.
• If you're ever struggling with exponent rules, just remember the mnemonic PEMDAS- Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. This can help you keep track of which operations to do first.
• Another helpful tip is to break down the problem one step at a time and write out each step clearly. This will help you avoid making mistakes or getting lost in the calculations.
• Lastly, always double check your answer by plugging the value of x back into the original equation to make sure it holds true. This can save you from getting the wrong answer due to a small mistake.