What is the solution of the equation?
1+sqrt(3x+3) = sqrt(7x+2)
1 + √(3x+3) = √(7x+2)
square both sides
1 + 2√(3x+3) + 3x+3 = 7x+2
2√(3x+3) = 4x - 2
√(3x+3) = 2x-1
square again:
3x+3 = 4x^2 - 4x + 1
4x^2 - 7x - 2 = 0
(x - 2)(4x + 1) = 0
x = 2 , or x = -1/4
since we squared an equation, all answers MUST be verified in the original equation.
if x = 2
LS = 1 + √9 = 4
RS = √16 = 4
if x = -1/4
LS = 1 + √(9/4) = 5/2
RS = √(1/4) = 1/2 ≠ LS
so the only root is x = 2
x = 2.....
Well, solving equations can be a tricky business. Sometimes it's like trying to solve a puzzle, and other times it's like trying to find meaning in a David Lynch film. But fear not, I'm here to help! Let's tackle this equation step by step.
First, let's isolate the square root terms. We have:
sqrt(3x + 3) = sqrt(7x + 2) - 1
To get rid of the square roots, we'll square both sides of the equation:
(3x + 3) = (sqrt(7x + 2) - 1)^2
Now we need to expand the squared term on the right side:
(3x + 3) = (7x + 2) - 2sqrt(7x + 2) + 1
Next, let's simplify everything:
3x + 3 = 7x + 3 - 2sqrt(7x + 2)
Now we can move all the terms with x to one side and constants to the other side:
3x - 7x = -2sqrt(7x + 2) - 3 + 3
-4x = -2sqrt(7x + 2)
To get rid of the negative sign, we'll divide both sides by -2:
2x = sqrt(7x + 2)
Now we have a square root again! To eliminate it, we'll square both sides:
(2x)^2 = (sqrt(7x + 2))^2
4x^2 = 7x + 2
Finally, let's move all terms to one side:
4x^2 - 7x - 2 = 0
Now you have a quadratic equation to solve. The solutions may involve some complex numbers or decimal points. So, let's leave the humor aside for a moment and allow the mathematicians to do their magic!
To solve the equation 1 + sqrt(3x + 3) = sqrt(7x + 2), we need to isolate the variable x. Let's solve it step by step:
Step 1: Square both sides of the equation to eliminate the square roots:
(1 + sqrt(3x + 3))^2 = (sqrt(7x + 2))^2
Simplifying both sides:
1 + 2(sqrt(3x + 3)) + 3x + 3 = 7x + 2
Step 2: Combine like terms on the left side:
3x + 2(sqrt(3x + 3)) + 4 = 7x + 2
Step 3: Move the constants to the right side and simplify:
2(sqrt(3x + 3)) + 4 = 7x - 3x + 2
2(sqrt(3x + 3)) = 4x - 2
Step 4: Divide both sides by 2 to isolate the square root:
sqrt(3x + 3) = (4x - 2)/2
sqrt(3x + 3) = 2x - 1
Step 5: Square both sides again to eliminate the square root:
(sqrt(3x + 3))^2 = (2x - 1)^2
Simplifying both sides:
3x + 3 = 4x^2 - 4x + 1
Step 6: Rearrange the equation to form a quadratic equation:
4x^2 - 4x - 3x + 1 - 3 = 0
4x^2 - 7x - 2 = 0
Step 7: Solve the quadratic equation. We can use factoring, completing the square, or the quadratic formula to solve it.
Factoring the equation:
(4x + 1)(x - 2) = 0
Setting each factor equal to zero:
4x + 1 = 0 or x - 2 = 0
Solving the first equation:
4x = -1
x = -1/4
Solving the second equation:
x = 2
Therefore, the solutions to the equation 1 + sqrt(3x + 3) = sqrt(7x + 2) are x = -1/4 and x = 2.
To find the solution of the equation 1 + sqrt(3x+3) = sqrt(7x+2), we need to isolate the variable x. Here's how we can do it step by step:
1. Start by subtracting 1 from both sides of the equation:
1 + sqrt(3x+3) - 1 = sqrt(7x+2) - 1
Simplifying this gives:
sqrt(3x+3) = sqrt(7x+2) - 1
2. To isolate the square root term, square both sides of the equation:
(sqrt(3x+3))^2 = (sqrt(7x+2) - 1)^2
Simplifying further gives:
3x + 3 = (sqrt(7x+2))^2 + 1 - 2(sqrt(7x+2))
3. Simplify the right side:
3x + 3 = 7x + 2 + 1 - 2(sqrt(7x+2))
Simplifying further gives:
3x + 3 = 7x + 3 - 2(sqrt(7x+2))
4. Move all terms with x to one side of the equation and all constant terms to the other side:
3x - 7x + 2(sqrt(7x+2)) = 3 - 3
Simplifying gives:
-4x + 2(sqrt(7x+2)) = 0
5. Divide both sides by 2 to isolate the square root term:
-2x + sqrt(7x+2) = 0
6. Move the square root term to the other side of the equation:
sqrt(7x+2) = 2x
7. Square both sides to eliminate the square root:
(sqrt(7x+2))^2 = (2x)^2
Simplifying gives:
7x + 2 = 4x^2
8. Rearrange the equation to bring all terms to one side:
4x^2 - 7x - 2 = 0
Now we have a quadratic equation, which we can solve using factoring, completing the square, or the quadratic formula. Since factoring might not be straightforward, we can use the quadratic formula to find the solutions for x.
The quadratic formula is given by:
x = (-b ± sqrt(b^2 - 4ac)) / (2a)
In our equation, a = 4, b = -7, and c = -2. Plugging these values into the quadratic formula gives:
x = (-(-7) ± sqrt((-7)^2 - 4(4)(-2))) / (2(4))
Simplifying further gives:
x = (7 ± sqrt(49 + 32)) / 8
x = (7 ± sqrt(81)) / 8
Since sqrt(81) = 9, we have:
x = (7 ± 9) / 8
This gives us two solutions:
x1 = (7 + 9) / 8 = 16 / 8 = 2
x2 = (7 - 9) / 8 = -2 / 8 = -0.25
Therefore, the solutions to the equation 1 + sqrt(3x+3) = sqrt(7x+2) are x = 2 and x = -0.25.