Math Problem Statement

If the sum and product of the roots of a quadratic equation are 8 and 15 respectively, then the equation is: (a) x2 − 8x + 15 = 0 (b) x2 + 8x + 15 = 0 (c) x2 − 15x + 8 = 0 (d) x2 + 15x − 8 = 0

Solution

The general form of a quadratic equation is:

ax2+bx+c=0ax^2 + bx + c = 0

From Vieta's formulas, for a quadratic equation ax2+bx+c=0ax^2 + bx + c = 0, the sum of the roots r1+r2r_1 + r_2 and the product of the roots r1×r2r_1 \times r_2 are given by:

r1+r2=baandr1×r2=car_1 + r_2 = -\frac{b}{a} \quad \text{and} \quad r_1 \times r_2 = \frac{c}{a}

For the given quadratic equation:

  • Sum of the roots = 8
  • Product of the roots = 15

Thus, we have:

r1+r2=8andr1×r2=15r_1 + r_2 = 8 \quad \text{and} \quad r_1 \times r_2 = 15

Comparing with Vieta's formulas, we know:

ba=8andca=15-\frac{b}{a} = 8 \quad \text{and} \quad \frac{c}{a} = 15

Assuming a=1a = 1 (since the coefficient of x2x^2 is not specified), we get:

b=8andc=15b = -8 \quad \text{and} \quad c = 15

Thus, the quadratic equation is:

x28x+15=0x^2 - 8x + 15 = 0

Therefore, the correct answer is:

(a) x28x+15=0x^2 - 8x + 15 = 0

Would you like a detailed explanation or have any other questions on this topic? Here are a few related questions for you:

  1. How can you use Vieta's formulas to find the roots of a quadratic equation?
  2. What happens to the quadratic equation if the product of the roots is negative?
  3. Can you derive the quadratic equation if the roots are given directly?
  4. How can you calculate the discriminant of a quadratic equation from the sum and product of its roots?
  5. What is the relationship between the sum and product of the roots and the vertex of the quadratic function?

Tip: Always remember that the sum and product of the roots can give you quick insights into the structure of a quadratic equation!

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Equations
Vieta's Formulas

Formulas

r1 + r2 = -b/a
r1 * r2 = c/a

Theorems

Vieta's Formulas

Suitable Grade Level

Grades 8-10