### Math problem #48

**Math problem:** It is known that and are the roots of the **quadratic equation**

Derive a quadratic equation whose roots are and .

**Discussion and hints:** Let us consider a general **quadratic equation**

Without loss of generality we may assume that To derive the formulas for the roots of the general quadratic equation (1) we complete the square on the left hand side to get

Introduce the * discriminant * of (1). It is clear from the formula above that if the quadratic equation (1) has two real roots, which are given by the

**quadratic formula**:

If then (1) admits one (double) real root

If then (1) has two

**complex-conjugate**roots

From the formulas that we have derived above, it is easy to compute (

**Viète's formulas**)

**Figure 1**

Let us also say a few words about maximum and minimum of the

**quadratic polynomial**function

If then the polynomial has no maximum value, but has the minimum of at If then the polynomial ha no minimum, but it has the maximum value of at

The graph of the function is a parabola. Its vertex is at the point

If the parabola is concave up, if the parabola is concave down. The vertical axis of such a parabola is the line

See

**Figure 1**for more details.

To complete our discussion let us consider the inequality defined by a quadratic polynomial:

If and () the inequality (3) holds true for any value of (see

**Figure 1**). If and () the inequality (3) is impossible for any value of ) (see

**Figure 1**). If and () the inequality (3) holds true for all values of satisfying or , where are the roots of the quadratic equation (1) (see

**Figure 1**). If and () the inequality (3) holds true for all values of satisfying , where are the roots of the quadratic equation (1) (see

**Figure 1**). If If and then the inequality (3) holds true for all values of If and (except , see

**Figure 1**). If If and then the inequality (3) is impossible for any value of and (see

**Figure 1**).

**Solution**

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