Find the inverse of the following quadratic equation. Hint: Complete the square first.
y= 2x²+16x-5
y = 2(x^2 + 8x +16-16) - 5
= 2( (x+4)^2 - 16) - 5
y = 2(x+4)^2 - 37
then the inverse is
x = 2(y+4)^2 - 37
so, does f(f-1(x)) = x?
can someone please help me A
16
ft
ladder leans against the side of a house. The bottom of the ladder is
9
ft
away from the side of the house. Find
x
, the angle of elevation of the ladder. Round your answer to the nearest tenth of a degree.
x
169
To find the inverse of a quadratic equation, we need to switch the roles of x and y, and then solve for y. Let's begin by completing the square on the given equation.
Step 1: Rearrange the equation:
y = 2x² + 16x - 5
Step 2: Move the constant term to the right side of the equation:
y + 5 = 2x² + 16x
Step 3: Factor out the common coefficient of x² and x:
y + 5 = 2(x² + 8x)
Step 4: Complete the square by adding the square of half of the coefficient of x to both sides of the equation:
y + 5 + (8/2)² = 2(x² + 8x + (8/2)²)
Simplifying this equation gives us:
y + 5 + 16 = 2(x + 4)²
Step 5: Simplify the equation:
y + 21 = 2(x + 4)²
Step 6: Divide both sides of the equation by 2 to isolate the squared term:
(y + 21) / 2 = (x + 4)²
Step 7: Take the square root of both sides to solve for x + 4:
±√[(y + 21)/2] = x + 4
Step 8: Subtract 4 from both sides:
x = ±√[(y + 21)/2] - 4
Step 9: Switch the roles of x and y:
y = ±√[(x + 21)/2] - 4
Therefore, the inverse of the quadratic equation y = 2x² + 16x - 5 is y = ±√[(x + 21)/2] - 4.