Solve in the exact form.
(sqrt of 4x+1)+(sqrt of x+1)=2
Someone showed me to do this next:
Square both sides..so..
4x+1+2((sqrt of 4x+1)•(sqrt of x+1))=4
I do not understand where the 2 come from ..and why do we need to multiply the sqrt of 4x+1 and sqrt of x+1 together to get the product.......??????????
Whoops I meant ..
Sqrt of x+3, not x+1..
(a+b)^2 = a^2 + ab + b^2
a+b)^2 = a^2 + 2ab + b^2
(sqrt of 4x+1)+(sqrt of x+3)=2
4x+1 + 2 (sqrt (x+3)sqrt(4x+1)) + x+3=4
5x+4 +2 (sqrt (5x+4)=4
5x+2sqrt(5x+4)=0
5x=-2(sqrt(5x+4)
square both sides
25x^2=4(5x+3)
25x^2-20x-12=0
(5x+2)(5x-6)=0
x=-12/5 or x=6/5
check that.
Hmmm. I get
√(4x+1)+√(x+3)=2
4x+1 + 2√(4x^2+13x+3) + x+3 = 4
5x = -2√(4x^2+13x+3)
25x^2 = 4(4x^2+13x+3)
25x^2 = 16x^2 + 52x + 12
9x^2 - 52x - 12 = 0
(9x+2)(x-6) = 0
x = -2/9 , 6
But x=6 does not satisfy the original equation, so -2/9 is the only solution.
To solve the equation (sqrt(4x+1)) + (sqrt(x+1)) = 2, the suggestion given was to square both sides of the equation. Let's break down each step to understand it:
Step 1: Square both sides of the equation:
(sqrt(4x+1))^2 + 2(sqrt(4x+1))(sqrt(x+1)) + (sqrt(x+1))^2 = 2^2
Step 2: Simplify the equation on each side:
4x + 1 + 2(sqrt(4x+1))(sqrt(x+1)) + x + 1 = 4
Step 3: Combine like terms:
5x + 2 + 2(sqrt(4x+1))(sqrt(x+1)) = 4
Now, let's address your specific questions:
1. Where does the 2 come from?
The 2 in the equation arises from squaring both sides of the original equation. When we square (sqrt(4x+1)) + (sqrt(x+1)), we obtain 2(sqrt(4x+1))(sqrt(x+1)) due to the nature of the multiplication.
2. Why do we need to multiply sqrt(4x+1) and sqrt(x+1) together to get the product?
Within the equation, we have a term with two square roots: (sqrt(4x+1))(sqrt(x+1)). This term represents the product of the square roots of (4x+1) and (x+1). Multiplying the square roots together allows us to combine them into a single expression in order to simplify the equation further.
By following these steps, we can continue solving the equation and find the exact form of the solution.