Question

Part A: Suppose the equivalent quadratic equation is written x2−18x+c=−80+c
. What value of c makes the equation a perfect square trinomial?

Part B: Factor the perfect square trinomial, and rewrite the equation as (x−h)2=k

Part C: Solve the quadratic equation and find a solution to x. Enter the lesser number.

Part D: Solve the quadratic equation and find a solution to x. Enter the greater number.

(4 points)

Answer 1

To solve your question step by step:

### Part A:
To write the quadratic equation \( x^2 - 18x + c = -80 + c \) in the form of a perfect square trinomial, we need to complete the square for the equation on the left side.

1. The left side of the equation is:
\[
x^2 - 18x + c
\]
2. To complete the square for \( x^2 - 18x \):
- Take half of the coefficient of \( x \) (which is -18), square it:
\[
\left(-\frac{18}{2}\right)^2 = (-9)^2 = 81
\]
3. So we can express the equation as:
\[
x^2 - 18x + 81
\]
4. Therefore, for \( c \) to make the left side a perfect square trinomial, we have:
\[
c = 81
\]

### Part B:
Now, we can factor the perfect square trinomial:
\[
x^2 - 18x + 81 = (x - 9)^2
\]
So we rewrite the equation as:
\[
(x - 9)^2 = -80 + 81
\]
This simplifies to:
\[
(x - 9)^2 = 1
\]

### Part C:
To find the solutions for \( x \):
1. We take the square root of both sides:
\[
x - 9 = \pm 1
\]
2. This gives us two equations:
- \( x - 9 = 1 \)
- \( x - 9 = -1 \)

From \( x - 9 = 1 \):
\[
x = 9 + 1 = 10
\]

From \( x - 9 = -1 \):
\[
x = 9 - 1 = 8
\]

The lesser number is \( \boxed{8} \).

### Part D:
The greater number from the solutions we found is \( \boxed{10} \).

In summary:
- Part A: \( c = 81 \)
- Part B: Equation is \( (x - 9)^2 = 1 \)
- Part C: Lesser solution \( x = 8 \)
- Part D: Greater solution \( x = 10 \)

Answer 2

8. An equation is shown. x2+4x+3=0
. Complete the statements.(4 points)
The value of a is
The value of b is
The value of c is
One solution to the equation is

Answer 3

Given the equation

\[
x^2 + 4x + 3 = 0
\]

we can identify the coefficients \(a\), \(b\), and \(c\) based on the standard form of a quadratic equation, which is

\[
ax^2 + bx + c = 0.
\]

### Completing the Statements:
1. **The value of** \(a\) **is** \(1\) (coefficient of \(x^2\)).
2. **The value of** \(b\) **is** \(4\) (coefficient of \(x\)).
3. **The value of** \(c\) **is** \(3\) (constant term).

### Finding One Solution to the Equation:
To find the solutions to the equation \(x^2 + 4x + 3 = 0\), we can factor it:

\[
(x + 1)(x + 3) = 0.
\]

Setting each factor to zero gives us:

1. \(x + 1 = 0 \quad \Rightarrow \quad x = -1\)
2. \(x + 3 = 0 \quad \Rightarrow \quad x = -3\)

Thus, one solution to the equation is

\[
\boxed{-1}.
\]

### Summary:
- The value of \(a\) is \(1\).
- The value of \(b\) is \(4\).
- The value of \(c\) is \(3\).
- One solution to the equation is \(-1\).

Answer 4

Using the quadratic equation above, fill in for a, b, and c.

Answer 5

For the quadratic equation provided:

\[
x^2 + 4x + 3 = 0
\]

we can identify the coefficients \(a\), \(b\), and \(c\) as follows:

- **The value of \(a\)** is \(1\) (the coefficient in front of \(x^2\)).
- **The value of \(b\)** is \(4\) (the coefficient in front of \(x\)).
- **The value of \(c\)** is \(3\) (the constant term).

So to fill in the statements:

- The value of \(a\) is **1**.
- The value of \(b\) is **4**.
- The value of \(c\) is **3**.

Let me know if you need anything else!