The equation x^2+12x=73 has two solutions. The positive solution has the form (sqrta)-(b) for positive natural numbers a and b. What is a+b?
Thanks
To find the solutions of the equation x^2 + 12x = 73, we need to rearrange it into the quadratic standard form (ax^2 + bx + c = 0). In this case, c = 73, b = 12, and a = 1.
Now, to solve the quadratic equation, we can use the quadratic formula:
x = (-b +/- sqrt(b^2 - 4ac)) / 2a
Plugging in the values, we have:
x = (-(12) +/- sqrt((12)^2 - 4(1)(73))) / 2(1)
x = (-12 +/- sqrt(144 - 292)) / 2
x = (-12 +/- sqrt(-148)) / 2
Since we are looking for real solutions, we can stop here. The square root of a negative number does not have a real solution.
Hence, there are no real solutions to the equation x^2 + 12x = 73. Therefore, a + b = 0.
To find the positive solution of the given equation x^2 + 12x = 73, we need to solve the equation by rearranging it into the quadratic form.
Step 1: Bring 73 to the other side of the equation:
x^2 + 12x - 73 = 0
Step 2: To solve a quadratic equation, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Comparing the equation x^2 + 12x - 73 = 0 with the general quadratic equation ax^2 + bx + c = 0, we can identify that a = 1, b = 12, and c = -73.
Step 3: Substitute the values of a, b, and c into the quadratic formula:
x = (-12 ± √(12^2 - 4(1)(-73))) / (2(1))
Simplifying the equation further:
x = (-12 ± √(144 + 292)) / 2
x = (-12 ± √436) / 2
Step 4: Simplify the square root:
x = (-12 ± √(4 * 109)) / 2
x = (-12 ± 2√109) / 2
x = -6 ± √109
Since we are looking for the positive solution, we take only the positive value:
x = -6 + √109
Step 5: The positive solution is in the form √a - b, where a = 109 and b = 6.
Finally, a + b = 109 + 6 = 115.
Therefore, the value of a + b is 115.
x^2 + 12x = 73
let's complete the square, the easiest way to solve this particular one
x^2 + 12x + 36 = 73+36
(x + 6)^2 = 109
x + 6 = ± √109
x = -6 ± √109
the positive one looks like
√109 - 6
so a = 1, b = -6
a+b = -5