Question

4x^2+69=36x quadratic formula

Answer 1

can it be simplified any more

Answer 2

To solve the quadratic equation 4x^2 + 69 = 36x using the quadratic formula, we first need to rearrange the equation into standard quadratic form:

4x^2 - 36x + 69 = 0

Now we can use the quadratic formula, which states that for the quadratic equation ax^2 + bx + c = 0, the solutions are given by:

x = (-b ± √(b^2 - 4ac)) / 2a

In this case, a = 4, b = -36, and c = 69. Substituting these values into the formula, we get:

x = (36 ± √((-36)^2 - 4*4*69)) / 2*4
x = (36 ± √(1296 - 1104)) / 8
x = (36 ± √192) / 8
x = (36 ± 4√12) / 8
x = (9 ± √12) / 2

Therefore, the solutions to the equation 4x^2 + 69 = 36x are:

x = (9 + √12) / 2 and x = (9 - √12) / 2

Answer 3

Yes, the solutions can be simplified further by rationalizing the denominator.

To simplify (9 + √12) / 2, we first rationalize the denominator by multiplying both the numerator and denominator by the conjugate of the denominator:

(9 + √12) / 2 * (2 - √12) / (2 - √12)
= (18 - 9√12 + 2√12 - 12) / (4 - 12)
= (6 - 7√12) / -8
= -3/4 - (7/8)√12

Similarly, for (9 - √12) / 2, we rationalize the denominator:

(9 - √12) / 2 * (2 + √12) / (2 + √12)
= (18 + 9√12 - 2√12 - 12) / (4 + 12)
= (6 + 7√12) / 16
= 3/8 + 7/16√12

Therefore, the simplified solutions to the quadratic equation are:

x = -3/4 - (7/8)√12 and x = 3/8 + 7/16√12