Solving the Quadratic Equation: (x + 15)^2 - 10 = 0
In this article, we will solve the quadratic equation (x + 15)^2 - 10 = 0 and find the values of x that satisfy the equation.
Expanding the Equation
First, let's expand the equation using the binomial theorem:
(x + 15)^2 = x^2 + 30x + 225
So, the equation becomes:
x^2 + 30x + 225 - 10 = 0
Simplifying the equation, we get:
x^2 + 30x + 215 = 0
Factoring the Equation
Unfortunately, the equation cannot be factored easily. Therefore, we will use the quadratic formula to solve for x.
Quadratic Formula
The quadratic formula is:
x = (-b ± √(b^2 - 4ac)) / 2a
In our case, a = 1, b = 30, and c = 215.
x = (-(30) ± √((30)^2 - 4(1)(215))) / 2(1)
x = (-30 ± √(900 - 860)) / 2
x = (-30 ± √40) / 2
x = (-30 ± 2√10) / 2
Solving for x
Now, we have two possible values for x:
x = (-30 + 2√10) / 2
x = (-30 - 2√10) / 2
Simplifying the Solutions
We can simplify the solutions by dividing the numerator and denominator of each expression by 2:
x = -15 + √10
x = -15 - √10
Finding the Lesser and Greater Values of x
Comparing the two values of x, we can conclude that:
Lesser value of x: x = -15 - √10 ≈ -22.16
Greater value of x: x = -15 + √10 ≈ -7.84
Therefore, the values of x that satisfy the equation (x + 15)^2 - 10 = 0 are x ≈ -22.16 and x ≈ -7.84.