Roman C. answered • 05/01/13

Masters of Education Graduate with Mathematics Expertise

Some equations can be solved exactly.

1. Linear equation, ax+b=0

Subtract b from both sides to get ax = -b.

Divide both sides by a to get x = -b/a.

2. Quadratics, ax^{2} + bx + c = 0. You can sometimes use factoring to solve it.

Otherwise you can use the quadratic formula.

3. Cubic and quartic equations.

You can always solve them exactly by radicals, but the general formulas are huge.

See my two video lessons on how to solve general cubic equations.

4. Some non-polynomial equations are solvable exactly.

ln(x+4) - 2 ln(x) - 1 = 0

ln(x+4) - ln(x^{2}) = 1

ln[(x+4)/x^{2}] = 1

(x+4) / x^{2} = e

x + 4 = ex^{2}

ex^{2} - x - 4 = 0

x = [1 + √(1+16e)] / (2e) The other root is discarded because it is negative, making ln(x) not real.

Some equations must be solved approximately with numerical algorithms, like Newton's method or Bisection Method. You can look these methods up at Wikipedia.

1. Polynomials of degree 5 or higher can't be solved exactly by radicals in general (Abel-Ruffini Theorem).

2. Many other equations can't be solved exactly, such as e^{x} - x - 2 = 0, or cos(x) - x = 0.

The solutions to the first equation are -1.84141... and 1.14619...

The solution to the second equation is 0.739085...