4x^2+4x=3
solve by completing the square
4x^2+4x=3
divide each term my 4
x^2 + x = 3/4
take 1/2 of the x-term coefficient, square that result, then add it to both sides
x^2 + x + 1/4 = 3/4 + 1/4
(x+1/2)^2 = 1
take root of both sides
x + 1/2 = ±1
x = -1/2 ± 1
x = -3/2 or 1/2
Well, it looks like we have a mathematical puzzle here. Time to put on my thinking wig... uh, I mean cap!
Let's start by rearranging the equation:
4x^2 + 4x = 3
4x^2 + 4x - 3 = 0
Now, to complete the square, we need to take half of the coefficient of x and then square it. The coefficient of x here is 4, so half of it is 2. Squaring 2 gives us 4, which means we need to add 4 to both sides of the equation:
4x^2 + 4x + 4 = 3 + 4
4x^2 + 4x + 4 = 7
Now, we can factor the left side as a perfect square:
(2x + 2)^2 = 7
Taking the square root of both sides:
2x + 2 = ±√7
Subtracting 2 from both sides:
2x = -2 ± √7
Finally, dividing both sides by 2:
x = (-2 ± √7)/2
Voila! The solutions to our puzzling equation are:
x = (-2 + √7)/2 or x = (-2 - √7)/2
I hope that puts a smile on your face!
To solve the quadratic equation 4x^2 + 4x = 3 by completing the square, follow these steps:
Step 1: Move the constant term to the right side of the equation:
4x^2 + 4x - 3 = 0
Step 2: Divide the entire equation by the coefficient of x^2 to make the leading coefficient equal to 1:
x^2 + x - 3/4 = 0
Step 3: Rewrite the equation by adding and subtracting the square of half of the coefficient of x:
x^2 + x + (1/2)^2 - (1/2)^2 - 3/4 = 0
Step 4: Simplify the equation:
(x + 1/2)^2 - 1/4 - 3/4 = 0
(x + 1/2)^2 - 4/4 = 0
(x + 1/2)^2 - 1 = 0
Step 5: Move the constant term to the right side of the equation:
(x + 1/2)^2 = 1
Step 6: Take the square root of both sides:
x + 1/2 = ±√1
Step 7: Simplify:
x + 1/2 = ±1
Step 8: Solve for x:
Case 1: x + 1/2 = 1
x = 1 - 1/2
x = 1/2
Case 2: x + 1/2 = -1
x = -1 - 1/2
x = -3/2
Therefore, the solutions to the given quadratic equation are x = 1/2 and x = -3/2.
To solve the equation 4x^2 + 4x = 3 by completing the square, follow these steps:
1. First, move the constant term (3) to the right side of the equation:
4x^2 + 4x - 3 = 0
2. Divide the entire equation by the coefficient of x^2 (4) to make the leading coefficient 1:
x^2 + x - 3/4 = 0
3. Complete the square for the quadratic expression x^2 + x. To do this, take half of the coefficient of x (1/2) and square it (1/4):
(x + 1/2)^2 - 1/4 - 3/4 = 0
Simplify the equation:
(x + 1/2)^2 - 4/4 = 0
(x + 1/2)^2 - 1 = 0
(x + 1/2)^2 = 1
4. Take the square root of both sides of the equation and consider both the positive and negative roots:
x + 1/2 = ± √1
x + 1/2 = ± 1
5. Solve for x by subtracting 1/2 from both sides of the equation:
x = -1/2 ± 1
So the solutions to the equation 4x^2 + 4x = 3 are:
x = -1/2 + 1
x = -1/2 - 1
Simplifying,
x = 1/2
x = -3/2