Question

1. the difference between the two numbers is 3.5. the product of those two numbers is 36. use a quadratic equation to find the numbers. show all work. (4 pts)

a. write system of equations: __________________________ (equation 1)
__________________________ (equation 2)
b. write the quadratic equation in term of one variable:
__________________________ (equation 3)
c. find 2 numbers:

answer(s): ___________________

Explanation:

Part a: System of Equations

Step1: Define variables

Let the two numbers be \( x \) and \( y \). Assume \( x > y \).

Step2: Equation for difference

The difference between the two numbers is 3.5, so \( x - y = 3.5 \) (Equation 1).

Step3: Equation for product

The product of the two numbers is 36, so \( xy = 36 \) (Equation 2).

Step1: Solve Equation 1 for \( x \)

From \( x - y = 3.5 \), we get \( x = y + 3.5 \).

Step2: Substitute into Equation 2

Substitute \( x = y + 3.5 \) into \( xy = 36 \):
\( (y + 3.5)y = 36 \)

Step3: Expand and simplify

Expand the left side: \( y^2 + 3.5y = 36 \)
Subtract 36 from both sides: \( y^2 + 3.5y - 36 = 0 \) (Equation 3)
(To eliminate decimals, multiply by 2: \( 2y^2 + 7y - 72 = 0 \), but the decimal form is also correct.)

Step1: Solve the quadratic equation

We have \( y^2 + 3.5y - 36 = 0 \). Multiply by 2 to eliminate decimals: \( 2y^2 + 7y - 72 = 0 \).
Use the quadratic formula \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 2 \), \( b = 7 \), \( c = -72 \).

Step2: Calculate discriminant

Discriminant \( D = b^2 - 4ac = 7^2 - 4(2)(-72) = 49 + 576 = 625 \).

Step3: Find roots

\( y = \frac{-7 \pm \sqrt{625}}{4} = \frac{-7 \pm 25}{4} \).
Two solutions:

• \( y = \frac{-7 + 25}{4} = \frac{18}{4} = 4.5 \)
• \( y = \frac{-7 - 25}{4} = \frac{-32}{4} = -8 \)

Step4: Find \( x \) for each \( y \)

• If \( y = 4.5 \), then \( x = y + 3.5 = 4.5 + 3.5 = 8 \).
• If \( y = -8 \), then \( x = -8 + 3.5 = -4.5 \). But check the product: \( (-4.5)(-8) = 36 \), which is valid, but typically we consider positive numbers unless specified.

(For positive numbers, \( y = 4.5 \), \( x = 8 \); or \( y = 8 \), \( x = 11.5 \)? Wait, no—wait, earlier we assumed \( x > y \), but let's recheck. Wait, if \( y = 4.5 \), \( x = 8 \): \( 8 - 4.5 = 3.5 \), and \( 8 \times 4.5 = 36 \). Correct. If \( y = -8 \), \( x = -4.5 \): \( -4.5 - (-8) = 3.5 \), and \( (-4.5)(-8) = 36 \). Also correct.)

Step5: Final numbers

The two numbers are \( 4.5 \) and \( 8 \) (or \( -4.5 \) and \( -8 \)).

Answer:

Equation 1: \( x - y = 3.5 \)
Equation 2: \( xy = 36 \)

Part b: Quadratic Equation in One Variable