How to Solve Complex Cases of Quadratic Equations

When a = 1
- Solving quadratic equations types x² + bx + c =
0.

Solving this type of quadratic equations results in solving a popular puzzle: finding two numbers knowing their sum and their product.

Solving becomes simple and doesn't need factoring.

Example
1.

Solve: x²
- 26x
- 72 =
0.

Solution.

Both real roots have opposite signs.

Write down the factor-pairs of c =
-72.

They are: (-1 , 72)(-2 , 36)(-3 , 24)...Stop!The sum of the 2 real roots in this set is 21 =
-b.

The 2 real roots are
-3 and
24.

Example
2.

Solve:
-x²
- 26x + 56 =
0.

Solution.

Roots have opposite signs.

Write down factor-pairs of c = 56: (-1, 56) (-2, 28)...Stop!.

This sum is 26 = b.

According to the Diagonal Sum Rule, when a is negative, the answers are
-2 and
28.

Example
3.

Solve x² + 27x + 50 =
0.

Solution.

Both real roots are negative.

Write factor-sets of c = 50: (-1,
-50) (-2,
-25)..Stop! This sum is
-27 =
-b.

The 2 real roots are
-2 and
-25.

Example
4.

Solve: x²
- 39x + 108 =
0.

Solution.

Both real roots are positive.

Write the factor-sets of c = 108: (1, 108) (2, 54) (3, 36)...Stop! This sum is 39 =
-b.

The 2 real roots are 3 and
36.;