Quadratic Equations

Step 1: Compute derivatives.



Step 2: Form new equation.



Step 3: Use condition that roots of are imaginary.
Imaginary roots means discriminant of is negative:

Step 4: Discriminant of new quadratic.



Step 5: Prove .
Given .
So is definitely negative.

Thus roots are imaginary.
Final Answer:

Correct Answer is Option 1: 10.8, 7.2

Step 1: Define the Variables

Let’s call the two numbers x and y. The problem gives us two key pieces of information:

• The sum of the numbers is 18, so: x + y = 18
• The ratio of their sum to their difference is 5:1. Assuming x > y, the difference is x - y. This gives us: (x + y) / (x - y) = 5

Step 2: Set Up the Equations

From the problem, we have two equations:

1. x + y = 18
2. (x + y) / (x - y) = 5

Since x + y = 18, substitute this into the second equation:

18 / (x - y) = 5

Step 3: Solve for the Difference

Cross-multiply to eliminate the fraction:

18 = 5(x - y)

Divide both sides by 5:

x - y = 18 / 5

x - y = 3.6

Now we have:

• x + y = 18
• x - y = 3.6

Step 4: Solve the System of Equations

Add the two equations to eliminate y:

(x + y) + (x - y) = 18 + 3.6

2x = 21.6

Divide by 2:

x = 21.6 / 2

x = 10.8

Substitute x = 10.8 into the first equation to find y:

10.8 + y = 18

y = 18 - 10.8

y = 7.2

Step 5: Verify the Solution

Let’s check if x = 10.8 and y = 7.2 satisfy the conditions:

• Sum: 10.8 + 7.2 = 18 (Correct!)
• Difference: 10.8 - 7.2 = 3.6
• Ratio: 18 / 3.6 = 5 (Correct, since 5 matches the 5:1 ratio!)

We are given the inequality:


Consider the critical points where expressions inside absolute values change sign:
and

Case 1:

So,

Case 2:

So,
Since , no solution in this interval.

Case 3:

So,

Combining all cases:

Final Solution Set: