Transforming and storing numbers with algebra

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• Using letters as placeholders in just like using variables in Python.

Solving First-Degree Equations

-One way to solve a simple equation like 2x+5 = 13 with programming is by using brute force.

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Finding the formula for first-degree equations

• Another way to solve an equation like 2x+5=13 is to find a general formula for this equation. We can then use this formula to write a program in Python. All first-degree equations fits into this general formula: ax + b = cx + d. On each side of the equal sign, you can see an x term and a constant, which is the number with no x attached to it. The number that precedes the x variable is called a coefficient.
• But sometimes there is no x term at all on one side of the equation, which means that the coefficient of that x is zero (cx = 0).

$$ ax + b = cx + d $$

$$ ax - cx = d - b $$

$$ x(a - c) = d - b $$

$$ x = \frac {d - b} {a - c} $$

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Writing the equation() function

• Write a function that takes four numbers a, b, c and d as parameters and plug in the general formula:

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Solving higher-degree equations

• Equations like x²+3x-10 = 0 are called quadratic equations and their general form is ax²+bx+c = 0, where a, b and c can be any number: positive or negative, whole numbers, fractions or decimals. The only exception is that a can't be 0 because that can make this a first-degree equation.
• Unlike first-degree equations which have one solution, quadratic equations have two possible solutions.
• When you isolate x:

$$ x = \frac{-b\sqrt{b²-4ac}}{2a} $$

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Using quad() to solve quadratic equations

• quad takes the three coefficients (a, b and c) and returns two solutions.
• We need to import the sqrt method from the math module. If you try to find the square of a negative number you'll see a value error.

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• The quad() function takes the numbers a, b and c as parameters and plugs them in to the quadratic formula.

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Using plug() to solve a cubic equation

• Cubic equations like 6x³+31x²+3x-10 = 0 have a term raised to the third degree.

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• We can solve one root using brute force, but the x³ term means there could be as many as three solutions to this equation. To see all the possible inputs and corresponding outputs we need a graph.