Solve equations reducible to quadratic equations
When solving equations, it is sometimes necessary to reduce them to quadratic equations - so that it's easier to solve.
Example 1.
In this equation, we have to look at (x^2 - 3x) as a single object/entity. This will make the equation easier to work with. Let (x^2 - 3x) be y. The equation then becomes y^2 - 4y + 3 = 0.
By factorising we get (y - 3)(y - 1) = 0. Therefore y = 3 or y = 1. x^2 - 3x = 3 or x^2 - 3x = 1.
1) x^2 - 3x - 3 = 0. By using the quadratic equation, we get: x = 3 +- root 21 all over 2
2) x^2 - 3x -1 = 0. By using the quadratic equation we get: x = 3 +- root 13 all over 2
By factorising we get (y - 3)(y - 1) = 0. Therefore y = 3 or y = 1. x^2 - 3x = 3 or x^2 - 3x = 1.
1) x^2 - 3x - 3 = 0. By using the quadratic equation, we get: x = 3 +- root 21 all over 2
2) x^2 - 3x -1 = 0. By using the quadratic equation we get: x = 3 +- root 13 all over 2
Example 2.
This can be viewed as a normal quadratic equation. By factorising, we get (x^2 - 4)(x^2 - 5) = 0
Therefore x = +-2 or x = +- root 5.
Therefore x = +-2 or x = +- root 5.