Solve:
1.) 2x² + 24 = x²
The answers I obtained were:
x = i√24 and x= -i√24 are these correct?
2.) 9x³ = 25x
3.) 5x² - x = 7x² - 5x - 2
4.) 2x² = x³ + 4x - 8
~Thank you
1. correct, I would change √24 to 2√6
2. 9x^3 - 25x = 0
x(9x^2 - 25) = 0
x(3x+5)(3x-5) = 0
x = 0, ±5/3
3. simplify to x^2 - 2x -1 = 0
then use the quadratic formula
4. x^3 - 2x^2 + 4x - 8 = 0
x^2(x-2) + 4(x-2) = 0
(x-2)(x^2+4) = 0
x = 2 or x = ±2i
You're welcome! Let's solve each of the equations step by step:
1.) 2x² + 24 = x²
To solve this equation, we can start by moving all the terms with x to one side of the equation and the constant terms to the other side.
Starting with the equation: 2x² + 24 = x²
Subtracting x² from both sides, we get:
2x² - x² + 24 = 0
This simplifies to: x² + 24 = 0
Now, we can solve this quadratic equation by factoring or using the quadratic formula. However, when we attempt to factor, we realize that it cannot be factored since there are no two numbers that multiply to give 24 and add up to 0.
Therefore, we will use the quadratic formula:
The quadratic formula is given by: x = (-b ± sqrt(b² - 4ac)) / (2a)
In our equation, a = 1, b = 0, and c = 24.
Substituting these values into the quadratic formula, we have:
x = (-0 ± sqrt(0² - 4(1)(24))) / (2(1))
Simplifying further:
x = (± sqrt(-96)) / 2
Taking the square root of a negative number gives an imaginary number. Therefore, the answers are:
x = ± i√96 or x = ± 4i√6
So, the answers you obtained, x = ± i√24, are not correct. The correct answers are x = ± 4i√6.
2.) 9x³ = 25x
To solve this equation, we can start by moving all the terms to one side:
9x³ - 25x = 0
Now, we can factor out x to simplify the equation:
x(9x² - 25) = 0
From the factored form, we can see that either x = 0, or 9x² - 25 = 0.
If we set 9x² - 25 = 0, we can solve for x by factoring the quadratic equation:
(3x - 5)(3x + 5) = 0
Setting each factor equal to zero, we have:
3x - 5 = 0 or 3x + 5 = 0
Solving these equations gives us:
x = 5/3 or x = -5/3
So, the solutions to the equation 9x³ = 25x are x = 0, x = 5/3, and x = -5/3.
3.) 5x² - x = 7x² - 5x - 2
To solve this equation, we start by moving all the terms to one side:
5x² - 7x² + x - 5x + 2 = 0
Combine like terms:
-2x² - 4x + 2 = 0
Now, we can try to factor the quadratic equation:
-2(x² + 2x - 1) = 0
However, it is not possible to factor this quadratic equation further.
At this point, we can solve for x using the quadratic formula:
x = (-b ± sqrt(b² - 4ac)) / (2a)
For our equation, a = -2, b = 2, and c = -1.
Substituting these values into the quadratic formula, we have:
x = (-2 ± sqrt(2² - 4(-2)(-1))) / (2(-2))
Simplifying further:
x = (-2 ± sqrt(4 - 8)) / (-4)
x = (-2 ± sqrt(-4)) / (-4)
Taking the square root of a negative number leads to imaginary solutions. Therefore, the solutions to the equation are complex numbers.
x = (-2 ± 2i) / (-4)
Simplifying further:
x = (1 ± i) / 2
So, the solutions to the equation 5x² - x = 7x² - 5x - 2 are x = (1 + i) / 2 and x = (1 - i) / 2.
4.) 2x² = x³ + 4x - 8
To solve this equation, we need to collect all terms on one side of the equation:
x³ - 2x² + 4x - 8 = 0
Now, we can try to factor the polynomial equation. However, its factors are not apparent.
In this case, we will need to use numerical approximation methods or a graphing calculator to find the solutions.