factor 9-6x-3x^2
The expression x^2-8x+k cannot be factored if k has the value
A.)7
B.)0
C)-7
D)-9
Please show me how to do these questions step by step so that I can understand & learn
9-6x-3x^2
notice the common factor of -3
= -3(x^2 + 2x - 3)
= -3(x + 3)(x - 1)
x^2 - 8x + k
If we have 1x^2 at the front, then the sum of the roots equal the coefficients of the middle term, and the product of the constants.
so x^2 - 8x +7 ??
two numbers that have a sum of -8 and a product of -7 are -1 and -7 , so it will factor
x^2 - 8x + 0 = x(x-8) , also works
x^2 - 8x - 7 ??
mmmh, can't thing of two numbers that add to -8 and multiply to get -7
x^2 - 8x - 9 = (x-9)(x+1)
So, what's the verdict ?
Oh I get it! It's -7! OMG thank u so much I really appreciate it :)
To factor the expression 9-6x-3x^2, follow these steps:
Step 1: Rearrange the terms in descending order based on the powers of x:
-3x^2 - 6x + 9
Step 2: Factor out the greatest common factor (GCF):
-The GCF of the coefficients -3, -6, and 9 is 3:
3(-x^2 - 2x + 3)
Step 3: Factor the quadratic expression within the parentheses:
- To factor the quadratic expression -x^2 - 2x + 3, we need to find two numbers that multiply to give 3 (the product of the coefficient of x^2 and the constant term) and add up to -2 (the coefficient of x). In this case, the numbers are -1 and -3:
- So, we can write the quadratic expression as:
3(-(x^2 + x - 3))
Step 4: Further factor the quadratic expression:
- Again, to factor the quadratic expression x^2 + x - 3, we need to find two numbers whose product is -3 (the constant term) and their sum is 1 (the coefficient of x). The numbers are 3 and -1:
- Rewrite the quadratic expression as:
3((x + 3)(x - 1))
So, the factored form of the expression 9-6x-3x^2 is:
3(x + 3)(x - 1)
For the second question, to determine the value of k for which the expression x^2 - 8x + k cannot be factored, we need to examine its discriminant.
The discriminant (Δ) of a quadratic equation in the form ax^2 + bx + c is calculated as Δ = b^2 - 4ac.
In our expression x^2 - 8x + k, the values for a, b, and c are 1, -8, and k, respectively.
By substituting these values into the discriminant formula, we get:
Δ = (-8)^2 - 4(1)(k)
= 64 - 4k
Now, we need to determine the values of k that make the discriminant less than zero, which means that the quadratic expression cannot be factored.
For the discriminant to be less than zero, we set 64 - 4k < 0 and solve for k:
64 - 4k < 0
-4k < -64
Divide both sides of the inequality by -4, remembering to reverse the inequality when dividing by a negative number:
k > 16
Therefore, any value of k greater than 16 will make the quadratic expression x^2 - 8x + k unable to be factored.
None of the answer choices A.) 7, B.) 0, C.) -7, or D.) -9 satisfy the condition, so the correct answer is none of the above.