(1) Sum and product of the roots

For the quadratic equation
\( x^2 - 7x + 6 = 0 \)
determine the sum and product of its roots using Viète's formulas, and then verify whether both roots can be natural numbers.

(2) Constructing an equation from given roots

The roots of a quadratic equation are
\( x_1 = -8 \) and \( x_2 = 5 \).
Construct a quadratic equation with a leading coefficient of 1 that has these roots and write it in standard form.

(3) Unknown root

One root of the equation
\( x^2 - 14x + 15 = 0 \)
is equal to \( 20 \). Determine the other root and verify the solution by applying Viète's formulas.

(4) Determination of a parameter

For which value of the parameter \( m \) does the equation
\( x^2 - (m + 7)x + m\cdot 3 = 0 \)
have roots whose sum is equal to \( 5 \) and whose product is equal to \( 1 \)?

(5) Quadratic equation with reciprocal roots

Determine the value of the number \( p \) such that the equation
\( x^2 - 10x + p = 0 \)
has roots that are reciprocal numbers. Afterwards, calculate the roots themselves.

(6) Equation from the sum of squares of the roots

The roots of a quadratic equation satisfy the conditions
\( x_1 + x_2 = 16 \) and \( x_1^2 + x_2^2 = 13 \).
Determine the product of the roots and then construct the corresponding quadratic equation with a leading coefficient of 1.

(7) Comparison of roots without calculating the discriminant

Consider the equation
\( x^2 - 12x + 8 = 0 \).
Using only Viète's formulas, determine whether both roots can be positive and less than the number \( 11 \).

(8) Integer roots

Determine all values of the number \( q \) for which the equation
\( x^2 - 12x + q = 0 \)
has two distinct integer roots. Write all possible pairs of roots.

(9) Equation whose roots are shifted by the same number

The roots of a certain quadratic equation are the numbers
\( -3 + -2 \) and \( 1 + -2 \).
Construct a quadratic equation with a leading coefficient of 1 and calculate the sum and product of its roots.

(10) Expression with the roots

If \( x_1 \) and \( x_2 \) are the roots of the equation
\( x^2 - 18x + 19 = 0 \),
calculate the value of the expression
\( x_1^3 + x_2^3 \)
without directly solving the equation, using only Viète's formulas.

(11) Condition on the squares of the solutions

In the equation
\( x^2 - (1 + m)x + m = 0 \)
determine the real number \( m \) knowing that its solutions \( x_1, x_2 \) satisfy the equality
\( x_1^2 + x_2^2 = 10 \).

(12) Determine the coefficients

Determine the coefficients \( p \) and \( q \) of the quadratic equation
\( x^2 + px + q = 0 \)
such that its solutions satisfy
\( x_1 = 1 \) and \( x_2 = 2 \).
Apply Viète's formulas to determine the coefficients.

(13) Simplify the fraction

\( \dfrac{x^2 - x - 6}{x^2 - 3x - 10} \).
Factorize the numerator and the denominator using known methods for quadratic expressions, and then write the final simplified form of the fraction.

(14) Value of an expression with the roots

If \( x_1 \) and \( x_2 \) are the solutions of the equation
\( x^2 - 4x - 1 = 0 \),
determine the value of the expression
\( \dfrac{x_1^2}{x_2} + \dfrac{x_2^2}{x_1} \)
using Viète's formulas without directly calculating the solutions of the equation.

(15) Determination of a parameter from one root

In the equation
\( x^2 - 9x + q = 0 \)
one solution is \( x_1 = 14 \), and the other solution is \( x_2 \).
Determine the value of the real parameter \( q \) using Viète's formulas.