To solve the quadratic equation \( x^2 - 18x = -80 \), we will first manipulate it and address each part step by step.
Part A: Find \( c \) that makes the equation a perfect square trinomial.
The original equation can be rewritten to find the value of \( c \): \[ x^2 - 18x + c = -80 + c \]
To complete the square, we take half of the coefficient of \( x \), which is \(-18\), divide it by 2 (which gives \(-9\)), and then square it: \[ (-9)^2 = 81 \]
Thus, \( c = 81 \) makes the expression a perfect square trinomial.
Answer for Part A: \[ c = 81 \]
Part B: Factor the perfect square trinomial and rewrite the equation as \( (x-h)^2 = k \).
Now, substituting \( c \) into the equation: \[ x^2 - 18x + 81 = -80 + 81 \] \[ x^2 - 18x + 81 = 1 \]
This can be factored as: \[ (x - 9)^2 = 1 \]
Answer for Part B: \[ (x - 9)^2 = 1 \]
Part C: Solve the quadratic equation and find the lesser solution for \( x \).
To solve the equation \( (x - 9)^2 = 1 \), we take the square root of both sides: \[ x - 9 = \pm 1 \]
This gives us two equations:
- \( x - 9 = 1 \) leads to \( x = 10 \)
- \( x - 9 = -1 \) leads to \( x = 8 \)
The lesser solution is: Answer for Part C: \[ x = 8 \]
Part D: Solve the quadratic equation and find the greater solution for \( x \).
From the above calculations, the greater solution is: Answer for Part D: \[ x = 10 \]
Final Summary of Answers:
Part A: \( c = 81 \)
Part B: \( (x - 9)^2 = 1 \)
Part C: \( x = 8 \)
Part D: \( x = 10 \)