Prove 3(x+1)(x+7)-(2x+5)² is never positive
So,
3(x+1)(x+7)-(2x+5)(2x+5)
=3(x²+8x+7)-(4x²+20x+25)
=3x²+24x+21-4x²-20x-25
=-x²+4x-4
= -(x^2-4x+4)
= -(x-2)^2
3(x + 1)(x + 7) - (2x + 5)(2x + 5)
= 3(x² + 8x + 7)- (4x²+ 20x + 25)
= 3x² + 24x + 21- 4x²- 20x- 25
= -x² + 4x - 4
= -1(x - 2)²
A squared number is always positive, but when you multiply it with a negative number, it is always negative, never positive.
^This will give you all 5 marks on Mathswatch. Hope it helps :)
Hi, you information was really useful but for some reason I only got 4 marks out of 5 on mathswatch, do you know why?
Sorry It actually did
this was extremely useful. thank you
THANK YOU SM
To prove that -x^2 + 4x - 4 is never positive, we can use algebraic manipulation and the concept of quadratic equations.
Let's solve the quadratic equation -x^2 + 4x - 4 = 0 to find the roots. If the roots are real, then we can determine whether the quadratic expression is never positive.
Using the quadratic formula, the roots of the equation are given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our quadratic equation -x^2 + 4x - 4 = 0, a = -1, b = 4, and c = -4.
Plugging in these values into the quadratic formula, we have:
x = (-(4) ± √((4)^2 - 4(-1)(-4))) / (2(-1))
x = (-4 ± √(16 - 16)) / (-2)
x = (-4 ± √(0)) / (-2)
Since the discriminant (denoted as b^2 - 4ac) is equal to 0, this means that the quadratic equation has repeated roots. In this case, that means the quadratic expression -x^2 + 4x - 4 = 0 has one real root, which is x = 2.
Now, let's consider the behavior of the quadratic expression -x^2 + 4x - 4 for values of x less than and greater than 2.
For x < 2:
Let's choose a point x = 1 and plug it into the quadratic expression:
(-1)(1^2) + 4(1) - 4 = -1 + 4 - 4 = -1
Since the value of the quadratic expression is negative for x < 2, it is not positive.
For x > 2:
Let's choose a point x = 3 and plug it into the quadratic expression:
(-1)(3^2) + 4(3) - 4 = -9 + 12 - 4 = -1
Again, we find that the value of the quadratic expression is negative for x > 2 and is not positive.
Therefore, we have shown that the quadratic expression -x^2 + 4x - 4 is never positive.
I don't know what you mean by it "didn't work." It just shows that the expression is never positive.
(x-2)^2 is a square, so it is always positive
So, -(x-2)^2 is never positive.
Trying to plug a formula into a request for a proof will never work. The logic is what counts.