Solve for x over the real numbers:4 x^4 - 37 x^2 + 9 = 0
Substitute y = x^2:4 y^2 - 37 y + 9 = 0
The left hand side factors into a product with two terms:(y - 9) (4 y - 1) = 0
Split into two equations:y - 9 = 0 or 4 y - 1 = 0
Add 9 to both sides:y = 9 or 4 y - 1 = 0
Substitute back for y = x^2:x^2 = 9 or 4 y - 1 = 0
Take the square root of both sides:x = 3 or x = -3 or 4 y - 1 = 0
Add 1 to both sides:x = 3 or x = -3 or 4 y = 1
Divide both sides by 4:x = 3 or x = -3 or y = 1/4
Substitute back for y = x^2:x = 3 or x = -3 or x^2 = 1/4
Take the square root of both sides:Answer(s): x = 3 or x = -3 or x = 1/2 or x = -1/2
For numbers 2-6, use inverse operations. Instead of subtracting add and if you are adding, subtract in the problem. For number 7, subtract $5 to $13. Hope this helps.
Let x be one number and y be the other.
#1 - Set up a system of equations.
4x = y
x + y = 45
#2 - Solve the system by substitution.
x + (4x) = 45
5x = 45
x = 9
(9) + y = 45
y = 36
Let's start from the left side, and manipulate it to get the right side, which would prove the identity.
We know that:
i)
ii)
.
From the above fundamental identities, we can rewrite the left hand side as:
,
which is what we needed to show.
If you see 8 more roses than lilacs, and there are 9 lilacs, then the answer would be 17 lilacs because 8 + 9 = 17. If you were to answer it through algebra, it would be 8 + 9 = x <span />
