Without using the root formula, find the roots of the quadratic equation x ^ 2-5x + 4 = 0, x ^ 2-6x-16 = 0.

1) Equation x ^ 2 – 5x + 4 = 0:

Let’s sum up all the coefficients of the equation:

1 + (-5) + 4 = 0.

A result of 0 means that one of the roots is 1.

The product of two roots of the equation is equal to a constant:

x1 * x2 = s.

Let’s find the second root from this formula:

1 * x2 = 4;

x2 = 4.

2) Equation x ^ 2 – 6x – 16 = 0.

Let’s sum up all the coefficients of the equation:

1 + (-6) + (-16) = -21.

The sum of two roots is equal to the coefficient b with the opposite sign:

x1 + x2 = -b.

We now have two formulas for the roots of the equation, which form a system of equations. Let’s find both roots from it:

x1 + x2 = – (-6);

x1 * x2 = -16.

You can solve the system by substituting x1 through x2, or you can pick up the roots in your head.

After selection, we get:

x1 = -2;

x2 = 8.

Answer: 1) x1 = 1; x2 = 4.2) x1 = -2; x2 = 8.