Part A: Suppose the equivalent quadratic equation is written x2−18x+c=−80+c

Question

. 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!