Problem Statement: Find the value of x2+y2x2+y2, given x+y=12x+y=12 and xy=32xy=32.

Solution:

Step 1: Understanding the problem

• We have two equations:
  1. x+y=12x+y=12
  2. xy=32xy=32
• We need to find the value of x2+y2x2+y2.

Step 2: Solving the equations

• We'll use the method of substitution to solve for xx and yy.
• From x+y=12x+y=12, we can express yy in terms of xx as y=12−xy=12−x.
• Substitute this expression for yy into equation 2: xy=32xy=32.
• We get x(12−x)=32x(12−x)=32.

Step 3: Finding the values of xx and yy

• Expanding the equation, we have 12x−x2=3212x−x2=32.
• Rearranging terms, we get x2−12x+32=0x2−12x+32=0.
• Now, we solve this quadratic equation for xx.
• We can use factoring or the quadratic formula to find the values of xx.
• Upon solving, we find two solutions for xx, let's call them x1x1 and x2x2.

Step 4: Finding corresponding values of yy

• Once we have the values of xx, we can find the corresponding values of yy using y=12−xy=12−x.

Step 5: Calculating x2+y2x2+y2

• For each pair of xx and yy, calculate x2+y2x2+y2.
• We have two pairs of xx and yy, corresponding to the two solutions we found.
• So, we calculate x12+y12x12+y12 and x22+y22x22+y22.

Step 6: Presenting the solution

• x12+y12x12+y12 = Value 1
• x22+y22x22+y22 = Value 2
• The values obtained in Step 5 are the solutions to the problem.

Final Answer:

• x2+y2=x2+y2= The sum of Value 1 and Value 2.

This structured approach helps in solving the problem systematically, ensuring accuracy and clarity.

Problem Analysis: Given equations:

1. 3x+2y=123x+2y=12
2. xy=6xy=6

We need to find the value of 9x2+4y29x2+4y2.

Solution:

Step 1: Find the values of xx and yy

To solve the system of equations, we can use substitution or elimination method.

From equation (2), xy=6xy=6, we can express yy in terms of xx: y=6xy=x6

Substitute this expression for yy into equation (1): 3x+2(6x)=123x+2(x6)=12

Now solve for xx:

3x+12x=123x+x12=12 3x2+12=12x3x2+12=12x 3x2−12x+12=03x2−12x+12=0

Divide the equation by 3: x2−4x+4=0x2−4x+4=0

Factorize: (x−2)2=0(x−2)2=0

So, x=2x=2.

Now, substitute x=2x=2 into equation (2) to find yy: 2y=62y=6 y=3y=3

So, x=2x=2 and y=3y=3.

Step 2: Find the value of 9x2+4y29x2+4y2

Substitute the values of xx and yy into the expression 9x2+4y29x2+4y2: 9(2)2+4(3)29(2)2+4(3)2 9(4)+4(9)9(4)+4(9) 36+3636+36 7272

Conclusion: The value of 9x2+4y29x2+4y2 is 7272.

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Heading 1: The Problem at Hand - Unraveling the Equation: Let's tackle the given mathematical problem: Compute the value of 9x² + 4y² if xy = 6 and 3x + 2y = 12. Our seasoned tutors at UrbanPro are well-equipped to guide students through such challenges.

Heading 2: Step-by-Step Solution - Navigating the Equation Maze: To solve the problem, we follow a systematic approach:

Sub-heading 1: Expressing y in terms of x:

• Utilize the given information, xy = 6.
• Substitute xy = 6 into the equation 3x + 2y = 12.
• Solve for y in terms of x.

Sub-heading 2: Substitution into the Expression:

• Once y is expressed in terms of x, substitute the value into the expression 9x² + 4y².
• This substitution simplifies the expression and facilitates further calculations.

Sub-heading 3: Simplification and Calculation:

• After substitution, simplify the expression and perform the necessary calculations.
• This step-by-step approach ensures accuracy and clarity in arriving at the final result.

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Perimeter Calculation for Rectangle with Given Area

Given Information:

• Area of the rectangle: 25x2−35x+1225x2−35x+12

Step 1: Determine the Dimensions

To calculate the perimeter of a rectangle, we need to know its length and width. We can find these dimensions using the area provided.

Step 2: Factorize the Area

Factorize the given quadratic expression 25x2−35x+1225x2−35x+12 to find its factors, which represent the possible lengths and widths of the rectangle.

Step 3: Use Factorization to Find Dimensions

Once the quadratic expression is factorized, identify the pairs of factors that, when multiplied, give the area of the rectangle. These pairs represent possible lengths and widths.

Step 4: Calculate Perimeter

With the length and width of the rectangle known, calculate the perimeter using the formula:

Perimeter=2×(Length+Width)Perimeter=2×(Length+Width)

Step 5: Finalize

Plug in the values of length and width into the perimeter formula to obtain the final result.

Let's proceed with these steps to find the perimeter.