To solve the problem, let's start by rewriting and analyzing the given quadratic equation.
The given quadratic equation is:
\[ x^2 - 18x = -80 \]
Part A: Finding the value of c
To convert the expression \( x^2 - 18x + c = -80 + c \) into a perfect square trinomial, we can complete the square. The term \( c \) can be determined using the formula:
\[ c = \left( \frac{b}{2} \right)^2 \]
where \( b \) is the coefficient of \( x \) (in this case, \( -18 \)).
Calculating \( c \):
\[ b = -18 \implies \frac{b}{2} = \frac{-18}{2} = -9 \] \[ c = (-9)^2 = 81 \]
Thus, the value of \( c \) that makes the equation a perfect square trinomial is 81.
Part B: Factor the perfect square trinomial
Insert the value of \( c \) back into the equation:
\[ x^2 - 18x + 81 = -80 + 81 \] \[ x^2 - 18x + 81 = 1 \]
The left side can be factored as:
\[ (x - 9)^2 = 1 \]
Part C: Solve the quadratic equation and find the lesser number
Now we need to solve \( (x - 9)^2 = 1 \). Taking the square root of both sides:
\[ x - 9 = \pm 1 \]
This leads to two equations:
- \( x - 9 = 1 \)
- \( x - 9 = -1 \)
Solving each equation gives:
- \( x = 10 \)
- \( x = 8 \)
Thus, the lesser number is 8.
Part D: Solve the quadratic equation and find the greater number
From the solutions above, the greater number is 10.
Summary of Answers:
- Part A: 81
- Part B: \((x - 9)^2 = 1\)
- Part C: 8
- Part D: 10
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