Solve the Quartic Equation: x⁴ - x³ = 2x²

Question

What is the value of x?

$x^4-x^3=2x^2$

00:00 Find X

00:03 Arrange the equation so that the right side is 0

00:12 Factor with the X squared term

00:23 Take out the common factor from parentheses

00:33 Looking for solution that zeros each factor in multiplication

00:40 This is one solution

00:43 Now let's find the second solution

00:48 Factor using trinomials, observe the coefficients

00:55 Want to find 2 numbers that sum to B (-1)

00:58 And their product equals C (-2)

01:01 These are the appropriate numbers, let's substitute in multiplication

01:09 Looking for solution that zeros each factor in multiplication

01:13 And this is the solution to the question

To solve the problem $x^4 - x^3 = 2x^2$ , let's proceed as follows:

Step 1: Set the equation to zero.
$x^4 - x^3 - 2x^2 = 0$
Step 2: Factor out the greatest common factor.
The common factor among all terms is $x^2$ .
Factoring out $x^2$ gives:
$x^2(x^2 - x - 2) = 0$
Step 3: Solve the factors.
This equation breaks into two factors that can be solved separately:
- $x^2 = 0$
- $x^2 - x - 2 = 0$
Step 4: Solve $x^2 = 0$ .
Since $x^2 = 0$ , we get:
$x = 0$
Step 5: Solve $x^2 - x - 2 = 0$ .
This can be factored further. We look for two numbers that multiply to $-2$ and add up to $-1$ .
These numbers are $-2$ and $1$ , so we factor as:
$(x - 2)(x + 1) = 0$
Step 6: Solve the quadratic factors.
Set each factor equal to zero:
- $x - 2 = 0 \Rightarrow x = 2$
- $x + 1 = 0 \Rightarrow x = -1$

The solutions to the equation $x^4 - x^3 = 2x^2$ are $x = -1, 0, 2$ .

Therefore, the correct answer is:

$x = -1, 2, 0$

$x=-1,2,0$