Question

If 2 ≤ 4x^2-9x + 20/x^2 +2 ≤ 3, x ∈ R, then x is ________. (The answer is [2, 7], please show how to solve)

Answer 1

Answer:

Quadratic Equations

A quadratic equation is one of the form ax2 + bx + c = 0, where a, b, and c are numbers, and a is not equal to 0.

Factoring

This approach to solving equations is based on the fact that if the product of two quantities is zero, then at least one of the quantities must be zero. In other words, if a*b = 0, then either a = 0, or b = 0, or both. For more on factoring polynomials, see the review section P.3 (p.26) of the text.

Example 1.

2x2 - 5x - 12 = 0.

(2x + 3)(x - 4) = 0.

2x + 3 = 0 or x - 4 = 0.

x = -3/2, or x = 4.

Square Root Principle

If x2 = k, then x = ± sqrt(k).

Example 2.

x2 - 9 = 0.

x2 = 9.

x = 3, or x = -3.

Example 3.

Example 4.

x2 + 7 = 0.

x2 = -7.

x = ± .

Note that = = , so the solutions are

x = ± , two complex numbers.

Completing the Square

The idea behind completing the square is to rewrite the equation in a form that allows us to apply the square root principle