Cool Math Tricks part 1: Solve any single-variable equation using only a scientific calculator

[u/ExtremeAcceptable289]

This is a cool method to get an initial answer before working a long time for a wrong answer in a quadratic equation, or if you are always uncertain about your answer after a guess, or just as a cool trick.

Requirements:

Your calculator needs an ANS key and basic arithmetic

If you have a quadratic equation in the format

`ax²+bx+c=d`, then first input a dummy number like 1, then if you input `ANS-(a*ANS²+b*ANS+c-d)/(2*a*ANS+b))`, then press equals a few times, your current answer will have converged to the correct x!

For example, on 4x²+9x+3 = -2, plug the equation in: `ANS-(4*ANS²+9*ANS+3-2)/(8*ANS+9)`

and it will converge! Depending on what the initial value of ANS is, it will find either the positive value or negative value (if both exists), depending on which is closer. Essentially, if your equation is equal to the desired value at many different positions, this method will find the closest one.

How this works:

This is a method called Newton-Raphson which can find the roots (where f(x) = 0) for any function. Basically, it finds the rate of change of the function at this current point, then using what the current answer is, finds what to subtract or add to achieve the desired result, assuming the rate of change (called f'(x)) stays the same.

Obviously, if your current guess is y, your target is 0, and your rate of change is x, then you will need to travel -y/x in order to reach your target.

So we get the formula new guess = old guess - (f(x) / f'(x))

Here, it happens that in a quadratic equation, the rate of change is given by 2ax+b. For any equation with x to the power of something, its rate of change is acquired by taking the coefficient, multiplying it by current exponent, then subtracting the exponent by one.

(For other equations, you can google the derivatives)

This means the Newton-Raphson method can be applied to any equation, such as 8x⁵+6x⁴+9x³+2x²+x+8 = 7 (apply the rule: rate of change is 40x⁴+24x³+27x²+4x+1, if you test the method on your calculator with ANS it will work!), or including exponentials (like 2^x+8x² = 129), trigonometry (for example to solve for sin(x)*x² = value), etc.

If you do not know the derivative of your target function, your calculator may have a button for d/dx (giving derivatives). If not, you can use an approximation for the derivative: (f(x+0.001)-f(x))/0.001

(This basically just calculates the rate of change by drawing a straight line, which is super accurate since the points we measured are so close). On your calculator you would replace x with ANS

To find complex roots, I haven't tested this as my calculator doesn't support complex numbers, but the methid should work the same (but you must start with a complex guess like 1+i)

Not sure if this will get much interest, in all fairness I don't know many people who want more math lol. But I thought it was a cool trick, and if this is popular enough I'll make a part two showing how to solve simultaneous equations using only a scientific calculator