To solve the quadratic equation (x+36)^2 - 38 = 251 using square roots, we can follow these steps:
Step 1: Expand the Equation
Expand the squared term (x+36)^2 to get: x^2 + 72x + 1296. The equation then becomes: x^2 + 72x + 1296 - 38 = 251.
Step 2: Simplify the Equation
Combine like terms to simplify the equation: x^2 + 72x + 1258 = 251.
Step 3: Move Constants to the Other Side
To isolate the squared term and x term, subtract 251 from both sides of the equation: x^2 + 72x + 1258 - 251 = 0. This results in: x^2 + 72x + 1007 = 0.
Step 4: Apply the Quadratic Formula
Using the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a), where a, b, and c are the coefficients of the quadratic equation (in the format ax^2 + bx + c = 0), we can find the solutions.
From the equation x^2 + 72x + 1007 = 0, we have:
a = 1 (coefficient of x^2),
b = 72 (coefficient of x),
c = 1007 (constant).
Plugging these values into the quadratic formula, we get:
x = (-72 ± √(72^2 - 4 * 1 * 1007)) / (2 * 1).
Simplifying the equation further:
x = (-72 ± √(5184 - 4028)) / 2.
x = (-72 ± √1156) / 2.
x = (-72 ± 34) / 2.
Step 5: Solve for x
Using the two different values of ±34, we can solve for x:
For the positive square root:
x = (-72 + 34) / 2 = -19.
For the negative square root:
x = (-72 - 34) / 2 = -53.
Therefore, the solutions to the quadratic equation (x+36)^2 - 38 = 251 using square roots are:
x = -19 and x = -53.