Question

Given a quadratic equation, $$ a x^{2}+b x+c=0, $$ the two roots are $$ x_{1}=\frac{-b+\sqrt{b^{2}-4 a c}}{2 a}, \quad x_{2}=\frac{-b-\sqrt{b^{2}-4 a c}}{2 a} $$ What are the problems with the following program? $$ \begin{aligned} &a=2 ; b=1 ; c=2 \\ &\text { from math import sqrt } \\ &q=\operatorname{sqrt}(b * b-4 * a * c) \\ &x 1=(-b+q) / 2 * a \\ &x 2=(-b-q) / 2 * a \\ &\text { print } x 1, x 2 \end{aligned} $$

Short answer

The program tries to calculate the square root of a negative discriminant, failing in real numbers. Also, division by \(2a\) must be correctly structured.

Step-by-step solution

Identify the equation components

The quadratic equation provided is \( ax^2 + bx + c = 0 \). For the given problem, we set \( a = 2 \), \( b = 1 \), and \( c = 2 \).

Check the discriminant

The formula for the discriminant in a quadratic equation is \( b^2 - 4ac \). Substituting the given values, we get \( 1^2 - 4 \times 2 \times 2 = 1 - 16 = -15 \). Since the discriminant is negative, the quadratic equation has no real roots.

Identify the Python program errors

The program attempts to calculate real roots using the formula, but the discriminant is negative. The 'sqrt' function from the math module cannot process negative numbers (it results in a math domain error). Therefore, the error lies in attempting to calculate the square root of a negative number affecting the line `q=sqrt(b*b-4*a*c)`.

Correct the formula placement

In the expressions for \( x_1 \) and \( x_2 \), there's a mistake in order of operations due to the multiplication by 2 and division. It should be \(( -b + q ) / ( 2 * a ) \) and \(( -b - q ) / ( 2 * a ) \), using proper parentheses to ensure the correct operation order.

Key concepts

Discriminant

The discriminant is a key component of solving quadratic equations. It is given by the formula \( b^2 - 4ac \). The discriminant helps us to determine the nature of the roots of the quadratic equation:

• If the discriminant is positive, there are two distinct real roots.
• If the discriminant is zero, there is one unique real root (or a repeated real root).
• If the discriminant is negative, there are no real roots. Instead, the roots are complex numbers.

In our problem, the discriminant turns out to be \(-15\) which is negative. This means the equation has two complex conjugate roots rather than real numbers. Understanding the discriminant can help anticipate the type of solution needed and avoid errors in calculations.

Complex Numbers

Complex numbers come into play when dealing with the square root of a negative number, such as in our problem with a discriminant of \(-15\). Complex numbers are expressed in the form \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. The fundamental unit of complex numbers \( i \) is defined as \( \sqrt{-1} \).

• When you have a negative discriminant, like \(-15\), you need to compute roots using complex numbers.
• Instead of using the usual `sqrt` function from Python's math library, which does not handle negatives, use the `cmath` module which supports complex square roots.
• The roots of the quadratic can then be computed as \(\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(\sqrt{b^2 - 4ac}\) results in a complex number.

Incorporating complex numbers allows us to fully describe all solutions to quadratic equations, even when they don't neatly fit into the realm of real numbers.

Python Programming

Python provides tools to solve quadratic equations, but they need to be used correctly. One common issue arises when handling negative discriminants with the `math` library, which only works with real numbers. Instead, Python's `cmath` module can handle complex numbers and should be used when discriminants are negative.
In our exercise code:

• Import `cmath` instead of `math` to handle complex square roots. Use `cmath.sqrt()` to compute the square root of a negative number.
• Ensure you use parentheses properly in your calculations involving the quadratic formula to avoid order of operations errors.
• Use parentheses thus: `(-b + q) / (2 * a)` rather than `(-b + q) / 2 * a` to correctly compute the roots.

By switching to `cmath` and correcting the formula errors, you can effectively solve quadratic equations with both real and complex roots in Python.

Mathematical Errors

Mistakes in solving quadratic equations can occur for a few reasons, including incorrect calculations or improper use of mathematical tools.
Let's address common errors:

• Neglecting the nature of the discriminant can lead to attempting to compute real roots when none exist, causing errors.
• Using the `math` library for negative discriminants results in math domain errors. Switching to `cmath` solves this issue.
• Order of operations mistakes, like improperly placing parentheses, can yield incorrect roots. Remember to properly encapsulate operations you want done first.

Attention to detail and understanding the mathematical principles behind your calculations can help you avoid these errors, ensuring accurate and reliable solutions.