Answer:
-1/6 - √2/2, -1/6 + √2/2
Explanation:
To solve the equation 36x^2 + 12x + 1 = 18 using the square root property, we need to rearrange the equation and isolate the variable x.
Start by subtracting 18 from both sides of the equation: 36x^2 + 12x + 1 - 18 = 0 Simplifying the left side: 36x^2 + 12x - 17 = 0
Now we can apply the square root property, which states that if a^2 = b, then a = ±√b.
Rewrite the equation in the form ax^2 + bx + c = 0: 36x^2 + 12x - 17 = 0
Let's solve for x using the square root property.
Calculate the discriminant (b^2 - 4ac) to determine the nature of the solutions:
a = 36, b = 12, c = -17 Discriminant = b^2 - 4ac = (12)^2 - 4(36)(-17) = 144 + 2448 = 2592
Since the discriminant is positive (2592 > 0), the equation has two distinct real solutions.
Apply the square root property:
x = (-b ± √(b^2 - 4ac)) / (2a)
Substitute the values of a, b, and c into the formula: x = (-12 ± √2592) / (2 * 36)
Simplify: x = (-12 ± √2592) / 72
We can simplify the square root of 2592: √2592 = √(36 * 72) = √36 * √72 = 6√72
Substitute this back into the equation: x = (-12 ± 6√72) / 72
Simplify the expression by dividing both the numerator and the denominator by their greatest common divisor, which is 6: x = -2 ± √72 / 12
Further simplify the expression by breaking down the square root: x = -2 ± √(36 * 2) / 12 = -2 ± √36 * √2 / 12 = -2 ± 6√2 / 12
Divide both the numerator and the denominator by 2: x = (-2 ± 6√2) / 12 = -1/6 ± √2/2
The solutions to the equation 36x^2 + 12x + 1 = 18 are:
x = -1/6 + √2/2, -1/6 - √2/2
Therefore, the solutions, listed from least to greatest, are:
-1/6 - √2/2, -1/6 + √2/2