Algebra

Equations

Common Transformations

If 𝑎𝑏=𝑐𝑑 then 𝑎+𝑏𝑎−𝑏=𝑐+𝑑𝑐−𝑑 If 𝑥+𝑦=𝑎𝑥−𝑦=𝑏} then {𝑥=12(𝑎+𝑏)𝑦=12(𝑎−𝑏) (𝑥+𝑦)²+(𝑥−𝑦)²=2(𝑥²+𝑦²) (𝑥+𝑦)²−(𝑥−𝑦)²=4𝑥𝑦 (𝑥+𝑦)²=(𝑥−𝑦)²+4𝑥𝑦 (𝑥−𝑦)²=(𝑥−𝑦)²−4𝑥𝑦

Examples

2𝑎²−𝑏²+𝑐²−𝑥²2𝑎²−𝑏²−𝑐²−𝑥²=3𝑎²−𝑏²+𝑐²−𝑑²3𝑎²−𝑏²−𝑐²−𝑑² 𝑐²−𝑑²2𝑎²−𝑏²=𝑐²−𝑑²3𝑎²−𝑏² 9(𝑐²−𝑥²)=4(𝑐²−𝑑²) 𝑥=5𝑐²+4𝑑²

Simplify a Compound Fraction

1𝑎²−𝑎𝑏+𝑏²+1𝑎²+𝑎𝑏+𝑏²1𝑎²−𝑎𝑏+𝑏²−1𝑎²+𝑎𝑏+𝑏² Multiply the numerator and denominator by the L.C.M. of all the smaller denominators. (𝑎²+𝑎𝑏+𝑏²)+(𝑎²−𝑎𝑏+𝑏²)(𝑎²+𝑎𝑏+𝑏²)−(𝑎²−𝑎𝑏+𝑏²)=𝑎²+𝑏²𝑎𝑏

Sources and References

https://archive.org/details/synopsis-of-elementary-results-in-pure-and-applied-mathematics-pdfdrive