x^2=7x+12=0
Solve the following quadratic equation by completing the square. Simplify the solutions and rationalize denominators, if necessary.
assuming a typo for
x^2+7x+12 = 0
x^2 + 7x + (7/2)^2 = -12 + (7/2)^2
(x + 7/2)^2 = 1/4
x + 7/2 = ± 1/2
x = -7/2 ± 1/2
To solve the quadratic equation x^2 = 7x + 12 = 0 by completing the square, follow these steps:
Step 1: Move the constant term to the other side of the equation:
x^2 - 7x - 12 = 0
Step 2: Take half of the coefficient of the x-term and square it:
Coefficient of x-term = -7
Half of -7 = -7/2
(-7/2)^2 = 49/4
Step 3: Rewrite the equation by adding and subtracting the value found in step 2:
x^2 - 7x + 49/4 - 49/4 - 12 = 0
Step 4: Group the first three terms and factor as a perfect square:
(x^2 - 7x + 49/4) - 49/4 - 12 = 0
(x - 7/2)^2 - 49/4 - 48/4 = 0
(x - 7/2)^2 - 97/4 = 0
Step 5: Simplify the equation:
(x - 7/2)^2 = 97/4
Step 6: Take the square root of both sides:
x - 7/2 = ± √(97/4)
Step 7: Add 7/2 to both sides:
x = 7/2 ± √(97/4)
Step 8: Simplify the solutions and rationalize denominators, if necessary:
x = (7 ± √97) / 2
Therefore, the solutions to the quadratic equation x^2 = 7x + 12 = 0, after completing the square, are x = (7 + √97) / 2 and x = (7 - √97) / 2.
To solve the quadratic equation x^2 + 7x + 12 = 0 by completing the square, follow these steps:
1. Move the constant term (12) to the other side of the equation:
x^2 + 7x = -12
2. Take the coefficient of the linear term (7), divide it by 2, and then square the result:
7/2 = 3.5, 3.5^2 = 12.25
3. Add the squared result to both sides of the equation:
x^2 + 7x + 12.25 = -12 + 12.25
(x + 3.5)^2 = 0.25
4. Take the square root of both sides:
√(x + 3.5)^2 = √0.25
x + 3.5 = ±0.5
5. Solve for x by subtracting 3.5 from both sides of the equation:
x = -3.5 ± 0.5
The solution for the quadratic equation x^2 + 7x + 12 = 0, after completing the square, is x = -3 or x = -4. To rationalize the denominators, which is not necessary in this case since both solutions are already rational, you would need to have a non-integer number in the denominator.