Quadratic Quest

Discover the Magic of Quadratic Equations!
The secret formula: x² - (sum)x + (product) = 0

Part 1: Find the Quadratic Equation

Given the sum and product of roots, we can find the quadratic equation using the formula above!

Problem (i): Sum = -9, Product = 20

Using the formula: x² - (sum)x + (product) = 0

Substitute the values: x² - (-9)x + 20 = 0

Simplify: x² + 9x + 20 = 0

Final Answer: x² + 9x + 20 = 0

Problem (ii): Sum = 5/3, Product = 4

Using the formula: x² - (sum)x + (product) = 0

Substitute the values: x² - (5/3)x + 4 = 0

Multiply all terms by 3 to eliminate fraction: 3x² - 5x + 12 = 0

Final Answer: 3x² - 5x + 12 = 0

Problem (iii): Sum = -3/2, Product = -1

Using the formula: x² - (sum)x + (product) = 0

Substitute the values: x² - (-3/2)x + (-1) = 0

Simplify: x² + (3/2)x - 1 = 0

Multiply all terms by 2: 2x² + 3x - 2 = 0

Final Answer: 2x² + 3x - 2 = 0

Problem (iv): Sum = −(2−a)², Product = (a + 5)²

Using the formula: x² - (sum)x + (product) = 0

Substitute the values: x² - [−(2−a)²]x + (a + 5)² = 0

Simplify: x² + (2−a)²x + (a + 5)² = 0

Final Answer: x² + (2−a)²x + (a + 5)² = 0

Part 2: Find Sum and Product of Roots

For any quadratic equation ax² + bx + c = 0:

Sum of roots = -b/a

Product of roots = c/a

Problem (i): x² + 3x −28 = 0

Here, a = 1, b = 3, c = -28

Sum of roots = -b/a = -3/1 = -3

Product of roots = c/a = -28/1 = -28

Final Answer: Sum = -3, Product = -28

Problem (ii): x² + 3x = 0

First, write in standard form: x² + 3x + 0 = 0

Here, a = 1, b = 3, c = 0

Sum of roots = -b/a = -3/1 = -3

Product of roots = c/a = 0/1 = 0

Final Answer: Sum = -3, Product = 0

Problem (iii): 3/a² + 1/a = 10

First, rewrite the equation:

Multiply all terms by a²: 3 + a = 10a²

Rearrange: 10a² - a - 3 = 0

Here, a = 10, b = -1, c = -3

Sum of roots = -b/a = -(-1)/10 = 1/10

Product of roots = c/a = -3/10

Final Answer: Sum = 1/10, Product = -3/10

Problem (iv): 3y² − y − 4 = 0

Here, a = 3, b = -1, c = -4

Sum of roots = -b/a = -(-1)/3 = 1/3

Product of roots = c/a = -4/3

Final Answer: Sum = 1/3, Product = -4/3

Try It Yourself!

Enter sum and product of roots to generate a quadratic equation: