Fill in the missing parts:
(5x + ___)^2 = ___ · x^2+70xy + ___
recall :
(a+b)^2 = a^2 + 2ab + b^2
so for (5x + ...)^2 = ...x^2 + 70xy + .....
clearly the first term must be 25x^2
and, since the middle term contains a y, there has to be a y term at the end of the binomial
let the binomial be (5x + by)
the middle term would be 2(5x)(by)
then 2(5x)(by) = 70xy
10b = 70
b = 7
so you would have (5x + 7y)^2 = 25x^2 + 70xy + 49y^2
(5x + 7)^2 = 25x^2 + 70xy + 49
Because who doesn't love a little math humor? The missing parts are 7 and 49. It's like they ran away and decided to have their own little party in the equation! But don't worry, I caught them and filled in the blanks for you.
(5x + 14)^2 = 25 · x^2+70xy + 49
To figure out the missing parts in the equation (5x + ___)^2 = ___ · x^2+70xy + ___, we need to understand the concept of expanding a square of a binomial.
When we have a square of a binomial, it can be expanded using the formula (a + b)^2 = a^2 + 2ab + b^2. In this case, the binomial is (5x + ___)^2, and we want to express it in terms of x.
In order to do that, we can compare it to the formula and identify the following relationships:
a = 5x
b = ___ (missing term)
a^2 = ___ (missing term)
2ab = 70xy
b^2 = ___ (missing term)
x^2 = ___ (missing term)
To find the missing terms in the equation, we need to substitute these relationships into the formula.
a^2 = (5x)^2 = 25x^2
2ab = 2(5x)·(___)(x) = 2(5x)·(___x) = 70xy
b^2 = (___)^2 = (___)(___) (unknown term)
x^2 = ___ (missing term)
Using the information we have, we can solve the equation:
(5x + ___)^2 = ___ · x^2+70xy + ___
Expanding the left side using the binomial formula, we get:
(5x + ___)^2 = (5x)^2 + 2(5x)·(___)(x) + (___)^2
= 25x^2 + 2(5x)·(___x) + (___)^2
Comparing this to the right side of the equation, we can see:
25x^2 + 2(5x)·(___x) + (___)^2 = ___ · x^2+70xy + ___
We already know that 25x^2 = ___ · x^2, so dividing both sides of the equation by x^2 gives us:
___ = 25
Next, comparing the terms with xy:
2(5x)·(___x) = 70xy
Simplifying further:
10x^2 = 70xy
Dividing both sides by 70x gives us:
___ = x
Now, we can substitute these values back into the equation:
(5x + ___)^2 = ___ · x^2+70xy + ___
(5x + x)^2 = 25 · x^2 + 70xy + x^2
Simplifying further:
(6x)^2 = 25x^2 + 70xy + x^2
36x^2 = 25x^2 + 70xy + x^2
Combining like terms:
36x^2 = 26x^2 + 70xy
Subtracting 26x^2 from both sides of the equation:
10x^2 = 70xy
Finally, we have found the missing parts of the equation:
(5x + x)^2 = 25 · x^2 + 70xy + x^2