Question

Теорема о корнях квадратного уравнения если a+b+c=0

Answer 1

если a+b+c=0, то х1=1, х2=с/а

Answer 2

Theorem on the Roots of a Quadratic Equation when a + b + c = 0

When the sum of the coefficients of a quadratic equation is equal to zero (a + b + c = 0), the roots of the quadratic equation can be determined using the following theorem.

Quadratic Equation: The general form of a quadratic equation is given by ax^2 + bx + c = 0, where a, b, and c are constants and x is the variable.

Theorem: The roots of the quadratic equation ax^2 + bx + c = 0, where a + b + c = 0, can be determined using the following relationships:

1. Sum of the Roots: The sum of the roots, denoted as x1 + x2, is equal to -b/a. 2. Product of the Roots: The product of the roots, denoted as x1 * x2, is equal to c/a.

Example Calculation:

Let's consider an example with specific values for a, b, and c to illustrate the application of the theorem.

Given: a = 4 b = -3 c = 1

Using the theorem, we can calculate the roots of the quadratic equation ax^2 + bx + c = 0, where a + b + c = 0.

Calculation: 1. Discriminant (D): D = b^2 - 4ac - D = (-3)^2 - 4*4*1 - D = 9 - 16 - D = -7

Since the discriminant (D) is less than 0, the roots are complex.

2. Sum and Product of the Roots: - Sum of the roots (x1 + x2) = -b/a = 3/4 - Product of the roots (x1 * x2) = c/a = 1/4

Therefore, the roots of the quadratic equation are complex and can be expressed as x1 = (3 + i)/4 and x2 = (3 - i)/4, where i is the imaginary unit.

This example demonstrates how the theorem can be applied to find the roots of a quadratic equation when the sum of the coefficients is equal to zero.