SOLVE BY FACTORING...THANK YOU
2X^3 - 5X^2 - 3X=0
Take out x as the common factor:
x(2x^2-5x-3)=0;
From here, you can just factorise normally or complete the square.
x(2x^2+x-6x-3)=0;
x(x(2x+1)-3(2x+1))=0;
x(2x+1)(x-3)=0;
x=-1/2,0,3
2 x³ - 5 x² - 3 x = 0
x ( 2 x² - 5 x - 3 ) = 0
Factor of 2 x² - 5 x - 3:
2 x² - 5 x - 3 = ( 2 x² + x ) - 6 x - 3 = x ( 2 x + 1 ) - ( 6 x + 3 ) =
x ( 2 x + 1 ) - 3 ( 2 x + 1 ) =
( 2 x + 1 ) ∙ x - ( 2 x + 1 ) ∙ 3 = ( 2 x + 1 ) ( x - 3 )
2 x³ - 5 x² - 3 x = x ( 2 x + 1 ) ( x - 3 )
Using the Zero Factor Principle: If a b = 0 then a = 0 or b = 0 or both a = 0 and b = 0
In this case x = 0 and ( 2 x + 1 ) ( x - 3 ) = 0
So one of solution is x = 0
Now solve 2 x + 1 = 0
2 x = - 1
x = - 1 / 2
Solve x - 3 = 0
x = 3
The solutions are:
x = - 1 / 2 , x = 0 and x = 3
To solve the equation 2x^3 - 5x^2 - 3x = 0 by factoring, we need to express the equation as a product of factors. Here are the steps to factor the equation:
Step 1: Arrange the equation in descending order of powers of x:
2x^3 - 5x^2 - 3x = 0
Step 2: Factor out the common factor, if any:
In this case, there is no common factor that can be factored out from all terms.
Step 3: Look for any special factoring patterns:
Unfortunately, there are no special factoring patterns apparent in this equation.
Step 4: Apply the Rational Root Theorem:
The Rational Root Theorem states that if a polynomial equation has a rational root, it will be of the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. The constant term is -3 and the leading coefficient is 2. The factors of -3 are ±1, ±3, and the factors of 2 are ±1, ±2. So, we can try substituting these values into the equation to see if any of them make it zero.
By testing different values for x, we find that x = 1 is a root of the equation.
Step 5: Use synthetic division or long division to divide the polynomial by (x - 1):
Performing synthetic division or long division, we can divide 2x^3 - 5x^2 - 3x by (x - 1), which yields:
2x^2 - 3x = 0
Step 6: Factor the resulting quadratic equation:
The resulting quadratic equation, 2x^2 - 3x = 0, can be further factored as:
x(2x - 3) = 0
Step 7: Set each factor equal to zero and solve for x:
We set x = 0 and 2x - 3 = 0 and solve for x:
For x = 0, we get: x = 0
For 2x - 3 = 0, we get: 2x = 3, x = 3/2
Therefore, the solutions to the equation 2x^3 - 5x^2 - 3x = 0 are:
x = 0, x = 3/2, and x = 1.