Solve in the exact form.
Original problem:
(sqrt of 4x+1)+(sqrt of x+1)=2
4x+1 + 2√(4x^2+13x+3) + x+3 = 4
Where does the 2 come from according to the equation above?? This 2:-----> 2√(4x^2+13x+3)
Barry
answered
the question is
¡Ô(4x+1) + ¡Ô(x+1) = 2
Square both sides
(¡Ô(4x+1) + ¡Ô(x+1))^2 = 2^2
still remember the identity
(a+b)^2 = a^2 + 2ab + b^2
let ¡Ô(4x+1) be a
¡Ô(x+1) be b
as a^2 + 2ab + b^2
we can derived the equation
4x+1 + 2¡Ô(4x^2+13x+3) + x+3 = 4
Reiny
answered
square both sides
(√(4x+1) )^2 = (2 - √(x+1) )^2
4x+1 = 4 - 4√(x+1) + x+1
4√(x+1) = 4 - 3x
square both sides again
16(x+1) = 16 - 24x + 9x^2
16x + 16 = 9x^2 - 24x + 16
9x^2 - 40x = 0
x(9x-40) = 0
x = 0 or x = 40/9
HOWEVER, since we squared the equation, all answers must be verfied in the original equation
√(4x+1) + √(x+1) = 2
if x = 0
LS = √(0+1) + √(0+1)
= 1+1 = 2
= RS
if x = 40/9
LS =√(160/9 + 1) + √(40/9 + 1)
= √(169/9) + √(49/9)
= 13/3 + 7/3
= 20/3
≠ RS
So the only solution is x = 0
check with Wolfram :
http://www.wolframalpha.com/input/?i=solve+√%284x%2B1%29+%3D+2+-+√%28x%2B1%29
Explain Bot
answered
(sqrt(4x + 1) + sqrt(x + 1) = 2)
The number 2 comes from the equation when simplifying and solving for the unknown variable. Let's go through the steps to solve the equation and understand where the 2 comes from:
Step 1: Square both sides of the equation to eliminate the square roots:
(sqrt(4x + 1) + sqrt(x + 1))^2 = 2^2
Simplifying the left side of the equation:
(4x + 1) + 2(sqrt(4x + 1))(sqrt(x + 1)) + (x + 1) = 4
Step 2: Combine like terms on the left side:
4x + 1 + 2√((4x + 1)(x + 1)) + x + 1 = 4
Step 3: Simplify the middle term by multiplying:
4x + 1 + 2√(4x^2 + 13x + 3) + x + 1 = 4
At this point, the equation has been simplified, and the 2 is still present in the middle term. The 2 in the equation originated from squaring the square root term (√(4x + 1))^2, which gave 2√((4x + 1)(x + 1)).
