1) 2x²-7x-15
2) 9x²+24x+16
Can you show me all the steps...i don't know how to do it...Thankyou!!:)
first: add 7x to each side. Then divide both sides by 9
second: subtract 9x+16 from each side. Then divide both sides by 15
we have to factor not solve
1. Y = 2x^2-7x-15 = 0.
Use the AC method of factoring:
A*C = 2*-15 = -30 = 1*-30 = 3*-10.
Choose the pair of factors whose sum = B(-7): 3, and -10.
2x^2 + (3x-10x) - 15 = 0.
Group the 4 terms into 2 factorable pairs:
(2x^2-10x) + (3x-15)
2x(x-5) + 3(x-5)
(x-5) is common to both terms. So we factor out(x-5):
(x-5)(2x+3) = 0
x-5 = 0
X = 5.
2x+3 = 0
2x = -3
X = -1.5.
Solution set: X = 5, and -1.5.
2. 9x^2+24x+16 = 0.
The given expression is a perfect square: 9x^2+24x+16 = (3x+4)^2 = 0.
(3x+4)^2 = 0
Take sgrt of both sides:
+-(3x+4) = 0
3x+4 = 0
3x = -4
X = -1 1/3 = -1.33333.
-3x-4 = 0
-3x = 4
X = -1 1/3 = -1.33333.
Solution X = -1.33333, and -1.33333.
Note: The given expression was set to zero, because the solutions are x-intercepts and Y = 0 at the x-intercepts.
9x^2+24x+16. Factor only.
9x^2+24x+16 = (3x+4)^2 = (3x+4)(3x+4).
Thankyou so much!!
Of course! I can walk you through the process of factoring each of these quadratic expressions.
1) 2x² - 7x - 15
To factor this quadratic expression, we're looking for two binomials in the form (px + q)(rx + s) where p, q, r, s are constants. These binomials will multiply together to give us the original quadratic expression.
First, we need to find two numbers, say a and b, whose product is equal to the product of the leading coefficient (2) and the constant term (-15), and whose sum is equal to the coefficient of x (-7).
In this case, the product of 2 and -15 is -30. We need to find two numbers whose product is -30 and whose sum is -7.
The numbers that satisfy these conditions are -10 and 3. So, we can rewrite the middle term (-7x) as the sum of -10x and 3x:
2x² - 10x + 3x - 15
Now, we group the terms:
(2x² - 10x) + (3x - 15)
We can now factor out the greatest common factor from each binomial:
2x(x - 5) + 3(x - 5)
Notice that we now have a common binomial factor of (x - 5). We can factor it out:
(x - 5)(2x + 3)
So the factored form of the quadratic expression 2x² - 7x - 15 is (x - 5)(2x + 3).
2) 9x² + 24x + 16
To factor this quadratic expression, we follow the same steps as before. We're looking for two numbers whose product is equal to the product of the leading coefficient (9) and the constant term (16), and whose sum is equal to the coefficient of x (24).
In this case, the product of 9 and 16 is 144. We need to find two numbers whose product is 144 and whose sum is 24.
The numbers that satisfy these conditions are 12 and 12. So, we can rewrite the middle term (24x) as the sum of 12x and 12x:
9x² + 12x + 12x + 16
Now, we group the terms:
(9x² + 12x) + (12x + 16)
The greatest common factor in the first binomial is 3x, and in the second binomial is 4:
3x(3x + 4) + 4(3x + 4)
Notice that we now have a common binomial factor of (3x + 4). We can factor it out:
(3x + 4)(3x + 4)
Since both binomials are the same, we can rewrite it as:
(3x + 4)²
So the factored form of the quadratic expression 9x² + 24x + 16 is (3x + 4)².
I hope this helps! Let me know if you have any further questions.