Find the inverse of the following quadratic equation. Hint: Complete the square first.
y= x²+14x+50
y = (x+7)^2 + 1
x+7 = ±√(y-1)
x = -7 ± √(y-1)
As you can see from the ±, there is no one inverse. You have to pick which branch of the parabola you want to use.
To find the inverse of the quadratic equation y = x² + 14x + 50, we need to follow a few steps. First, we'll complete the square to convert the equation into vertex form, and then we'll interchange x and y to obtain the inverse equation.
Step 1: Complete the Square
To complete the square, we add and subtract the square of half the coefficient of x. In this case, the coefficient of x is 14, so we have:
y = x² + 14x + 50
= (x² + 14x + ?) + 50 - ?
To find the missing term, we take half of the coefficient of x and square it:
(14/2)² = 7² = 49
So our equation becomes:
y = (x² + 14x + 49) - 49 + 50
= (x + 7)² + 1
Step 2: Interchanging x and y
To find the inverse, we need to interchange x and y in the equation:
x = (y + 7)² + 1
Step 3: Solve for y
Now we'll solve the equation for y. Start by subtracting 1 from both sides:
x - 1 = (y + 7)²
Next, take the square root of both sides, remembering to consider both positive and negative roots:
√(x - 1) = ± (y + 7)
To isolate y, we'll subtract 7 from both sides:
√(x - 1) - 7 = ± y
Since we want to find the inverse function, we only consider the positive square root:
y = √(x - 1) - 7
So the inverse function is given by y = √(x - 1) - 7.