Please, Completely factor 5c^2-24cd-5d^2
Thank so much
5c^2-24cd-5d^2
What methods have you learned?
What have you got so far?
hint: 24 = 5*5 - 1
To completely factor the expression 5c^2 - 24cd - 5d^2, we can follow these steps:
Step 1: Look for common factors.
In this case, we cannot find any common factors among the terms, so we move to the next step.
Step 2: Check for perfect square trinomials.
A perfect square trinomial is a trinomial that can be factored into the square of a binomial. To check if the given trinomial is a perfect square trinomial, we can compare it with the general form (a^2 + 2ab + b^2) or (a^2 - 2ab + b^2).
The given trinomial 5c^2 - 24cd - 5d^2 does not fit the pattern of a perfect square trinomial, so we move to the next step.
Step 3: Use the quadratic formula.
Since the given trinomial is not easily factorable, we can use the quadratic formula to find the roots (or zeros) of the equation and then write it in factored form.
The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac))/(2a)
In our case, the equation is in the form ax^2 + bx + c = 0, with a = 5, b = -24, and c = -5.
To find the roots, we can substitute these values into the quadratic formula:
c = (-(-24) ± √((-24)^2 - 4(5)(-5)))/(2(5))
Simplifying further:
c = (24 ± √(576 + 100))/(10)
c = (24 ± √(676))/(10)
c = (24 ± 26)/(10)
This gives us two possible solutions:
c = (24 + 26)/(10) = 5
c = (24 - 26)/(10) = -1/5
The roots of the quadratic equation are c = 5 and c = -1/5.
Step 4: Write the factored form.
Now that we have the roots, we can express the quadratic as a product of linear factors.
The factored form is given by:
5c^2 - 24cd - 5d^2 = (c - 5)(c + 1/5)
Therefore, the expression 5c^2 - 24cd - 5d^2 is completely factored as (c - 5)(c + 1/5).