Question

2x^3 - 5x^2 + 3x = 0

Answer 1

To solve the equation 2x^3 - 5x^2 + 3x = 0, you can factor it to find its roots. Here's how you can do it:

Start with the given equation:

2x^3 - 5x^2 + 3x = 0

Factor out the common term 'x' from each term:

x(2x^2 - 5x + 3) = 0

Now, you need to factor the quadratic expression 2x^2 - 5x + 3. It can be factored as follows:

(2x - 3)(x - 1) = 0

Now, you have factored the original equation into two parts:

x(2x - 3)(x - 1) = 0

Use the zero-product property, which states that if a product of factors equals zero, at least one of the factors must equal zero.

Set each factor equal to zero and solve for x:

a. x = 0

b. 2x - 3 = 0

c. x - 1 = 0

Solve for x in the second equation:

2x - 3 = 0

2x = 3

x = 3/2

Solve for x in the third equation:

x - 1 = 0

x = 1

So, the solutions for the equation 2x^3 - 5x^2 + 3x = 0 are x = 0, x = 3/2, and x = 1.

Methods that you can do.

Answer 2

x = 1 and x = 3/2

Explanation:

To solve the equation 2x^3 - 5x^2 + 3x = 0, you can factor it and then find the solutions:

First, factor out the common term, which is x:

x(2x^2 - 5x + 3) = 0

Now, focus on the quadratic expression in the parentheses:

2x^2 - 5x + 3

To factor the quadratic expression, you're looking for two numbers that multiply to 2 * 3 = 6 and add up to -5 (the coefficient of the linear term). The numbers that fit this criteria are -2 and -3 because (-2) * (-3) = 6, and (-2) + (-3) = -5.

So, you can factor the quadratic expression as:

2x^2 - 5x + 3 = 2x^2 - 2x - 3x + 3

Now, factor by grouping:

2x(x - 1) - 3(x - 1)

Notice that (x - 1) is a common factor in both terms, so you can factor that out:

(x - 1)(2x - 3) = 0

Now, you have factored the equation completely. To find the solutions, set each factor equal to zero:

x - 1 = 0

x = 1

2x - 3 = 0

2x = 3

x = 3/2

So, the solutions to the equation are x = 1 and x = 3/2.