So I am doing my algebra home work and I get stuck on these three problems.
x^2-19x+72 from there idk what to do because nothing multiplies up to 72 and adds up to -19.
Same for this one.
x^2-25x+48
And
x^2+9-90
(X+15) (x-6)
I am assuming you are factoring ?
for x^2 - 19x + 72 there are no factors with rational numbers
(I found the discriminant b^2 - 4ac to be 649, which is not a perfect square, thus no factors)
x^2 - 25x - 90
same thing, b^2-4ac = 985 , no factors
I assume a typo in the 3rd and you meant:
x^2 + 9x - 90
which does factor to (x+15)(x-6)
Let's start by addressing each problem step by step.
Problem 1: x^2 - 19x + 72
To factorize the quadratic expression x^2 - 19x + 72, we need to find two numbers that multiply to 72 and add up to -19. Since we don't immediately see such numbers, we can use a trial and error method. We start by listing all the factors of 72:
1 × 72 = 72
2 × 36 = 72
3 × 24 = 72
4 × 18 = 72
6 × 12 = 72
8 × 9 = 72
From these pairs, we need to find the pair that adds up to -19. In this case, the numbers are -3 and -24. By replacing the middle term (-19x) with these numbers, we can rewrite the quadratic expression as:
x^2 - 3x - 24x + 72
Now we group the terms and factor by grouping:
x(x - 3) - 24(x - 3)
Notice that we have common factors (x - 3) that we can factor out:
(x - 3)(x - 24)
Therefore, the factored form of x^2 - 19x + 72 is (x - 3)(x - 24).
Problem 2: x^2 - 25x + 48
To factorize x^2 - 25x + 48, again we need to find two numbers that multiply to 48 and add up to -25. Using the trial and error method, we find that the numbers are -3 and -16. The expression can then be rewritten as:
x^2 - 3x - 16x + 48
By grouping, we can factor out common factors:
x(x - 3) - 16(x - 3)
Once again, we can factor out the common factor (x - 3):
(x - 3)(x - 16)
So the factored form of x^2 - 25x + 48 is (x - 3)(x - 16).
Problem 3: x^2 + 9x - 90
In this problem, we need to factorize the quadratic expression x^2 + 9x - 90. Since we already have three terms, we can directly look for two numbers that multiply to -90 and add up to 9. Using trial and error, we find the numbers 15 and -6. So the expression can be rewritten as:
x^2 + 15x - 6x - 90
By grouping, we can factor out common factors:
x(x + 15) - 6(x + 15)
Again, we can factor out the common factor (x + 15):
(x + 15)(x - 6)
Therefore, the factored form of x^2 + 9x - 90 is (x + 15)(x - 6).
Remember, factoring quadratic expressions often requires trial and error or other factoring techniques. Practicing and becoming familiar with different factoring methods will help you solve these types of problems more easily.