Quadratic Equations Problem Solving

Quadratic Equations: Problem Solving

• A rectangular table has an area of 27 square feet and a perimeter of 24 feet.
• Area formula: Length × Width = 27.
• Perimeter formula: 2(Length + Width) = 24.
• Simplifying perimeter equation results in Length + Width = 12.
• Set up a quadratic equation using the relationships: x² - 12x + 27 = 0.
• Factoring gives (x - 3)(x - 9) = 0, leading to roots x = 3 and x = 9.
• Width = 3 feet, Length = 9 feet (length is always the larger value).
• Confirmation: Area 3 × 9 = 27 square feet and Perimeter (2 × 3) + (2 × 9) = 24 feet.

Sum of Two Numbers Problem

• Two numbers have a sum of 19 and sum of their squares is 193.
• Let one number be x, the other is 19 - x.
• Representing the sum of squares: x² + (19 - x)² = 193.
• Expanding gives: x² + 361 - 38x + x² = 193.
• Combined terms yield: 2x² - 38x + 168 = 0.
• Dividing by 2 reduces to: x² - 19x + 84 = 0.
• Factoring results in (x - 7)(x - 12) = 0, giving roots x = 7 and x = 12.
• Therefore, the two numbers are 7 and 12.
• Verification: 7 + 12 = 19 and squares 49 + 144 = 193.

Inclined Ramp Problem

• Inclined ramp's length (hypotenuse) is 8 meters longer than the rise, and the run is 7 meters longer than the rise.
• Let x be the rise, making run = x + 7 and length = x + 8.
• Use the Pythagorean theorem: (x + 8)² = x² + (x + 7)².
• Expanding both sides yields: x² + 16x + 64 = x² + 14x + 49.
• Rearranging leads to x² - 2x - 15 = 0.
• Factoring gives: (x - 5)(x + 3) = 0, leading to valid solution x = 5 (length must be non-negative).
• Thus, Rise = 5 meters, Run = 12 meters, and Length (hypotenuse) = 13 meters.

Quadratic Equations: Problem Solving

• A rectangular table's area is 27 square feet.
• The perimeter of the table is 24 feet.
• Area can be calculated using the formula: Length × Width = 27.
• Perimeter is calculated with: 2(Length + Width) = 24, simplifying to Length + Width = 12.
• Set up the quadratic equation: x² - 12x + 27 = 0.
• The equation factors to (x - 3)(x - 9) = 0, resulting in roots: x = 3 and x = 9.
• Assign Width = 3 feet and Length = 9 feet; the larger value corresponds to length.
• Verification shows: Area calculation 3 × 9 = 27 square feet and Perimeter calculation (2 × 3) + (2 × 9) = 24 feet.

Sum of Two Numbers Problem

• The two numbers have a combined sum of 19.
• Their squared sum is indicated as 193.
• Define one number as x and the other as 19 - x.
• Establish the equation for the sum of squares: x² + (19 - x)² = 193.
• Expanding provides: x² + 361 - 38x + x² = 193.
• Combining yields the equation: 2x² - 38x + 168 = 0.
• This is reduced to: x² - 19x + 84 = 0 after dividing by 2.
• Factoring results in (x - 7)(x - 12) = 0, giving roots x = 7 and x = 12.
• The actual numbers are 7 and 12.
• Validation shows: 7 + 12 = 19 and 49 + 144 = 193.

Inclined Ramp Problem

• The inclined ramp's length (hypotenuse) is 8 meters longer than the rise.
• The run is defined as 7 meters longer than the rise.
• Let the rise be represented as x; then run = x + 7, length = x + 8.
• Apply the Pythagorean theorem: (x + 8)² = x² + (x + 7)².
• Expansion leads to the equation: x² + 16x + 64 = x² + 14x + 49.
• Rearranging results in the quadratic: x² - 2x - 15 = 0.
• Factoring the equation gives: (x - 5)(x + 3) = 0, with a valid solution of x = 5.
• The Rise is determined to be 5 meters, leading to a Run of 12 meters and a Length (hypotenuse) of 13 meters.

This quiz explores problem-solving techniques using quadratic equations, focusing on scenarios such as determining the dimensions of a rectangular table and solving for two numbers based on given conditions. Each question involves setting up and factoring quadratic equations to find solutions. Test your understanding and skills in applying quadratic formulas!