Question

Решение квадратичной функции (ax^2 + bx + c = 0) на Паскале ABC через дискриминант

Answer 1

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Answer 2

Solving a Quadratic Function Using the Discriminant in Pascal's ABC

To solve a quadratic function of the form \(ax^2 + bx + c = 0\) in Pascal's ABC using the discriminant, we can follow these steps:

1. Calculate the Discriminant (D): The discriminant, denoted as \(D\), is calculated using the formula: \[D = b^2 - 4ac\]

This discriminant is a key factor in determining the nature of the roots of the quadratic equation.

2. Determine the Nature of the Roots: - If \(D > 0\), the quadratic equation has two distinct real roots. - If \(D = 0\), the quadratic equation has two equal real roots. - If \(D < 0\), the quadratic equation has two complex roots.

3. Calculate the Roots: - If the discriminant is positive, the roots can be calculated using the quadratic formula: \[x = \frac{{-b \pm \sqrt{D}}}{{2a}}\] where \(D\) is the discriminant.

- If the discriminant is zero, the roots are real and equal, and can be calculated using: \[x = \frac{{-b}}{{2a}}\]

- If the discriminant is negative, the roots are complex and can be expressed in the form: \[x = \frac{{-b \pm i\sqrt{|D|}}}{{2a}}\]

Example Calculation Using the Given Coefficients

Let's consider an example with the coefficients provided in the search snippet:

Given coefficients: \(a = 1\), \(b = -7\), \(c = -10\)

1. Calculate the Discriminant: \[D = (-7)^2 - 4(1)(-10) = 49 + 40 = 89\]

2. Determine the Nature of the Roots: Since \(D = 89 > 0\), the quadratic equation has two distinct real roots.

3. Calculate the Roots: Using the quadratic formula: \[x = \frac{{-(-7) \pm \sqrt{89}}}{{2(1)}}\] \[x = \frac{{7 \pm \sqrt{89}}}{{2}}\]

Therefore, the roots of the quadratic equation \(x^2 - 7x - 10 = 0\) are: \[x_1 = \frac{{7 + \sqrt{89}}}{{2}}\] \[x_2 = \frac{{7 - \sqrt{89}}}{{2}}\]

This demonstrates how to solve a quadratic function using the discriminant in Pascal's ABC.