Solve the Quartic Equation: x⁴ - x³ = 2x²

Question

What is the value of x?

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

00:07 Let's find the value of X.

00:10 First, arrange the equation so the right side equals zero.

00:19 Next, factor the equation, starting with the X squared term.

00:30 Now, take out the common factor from the parentheses.

00:40 We're looking for solutions that make each factor equal zero.

00:47 Great! This gives us one solution.

00:51 Now, let's find the second solution.

00:55 Factor using trinomials. Pay attention to the coefficients.

01:02 We need two numbers that add up to B, which is negative one.

01:07 And their product should equal C, which is negative two.

01:12 These numbers fit perfectly. Let's substitute them into our multiplication.

01:18 Again, find solutions that make each factor zero.

01:22 And that's how we solve the problem. Well done!

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:

$x = -1, 2, 0$

$x=-1,2,0$