( Quadratic Inequalities

Quadratic Inequalities

Consider the inequality:

x2 − 6x + 8 > 0

It can be solved by following these steps:

1.Equal the polynomial of the first member to zero and the roots of the quadratic equation are obtained.

x2 − 6x + 8 = 0

Quadratic Inequality


2. Represent these values in the real line. Take one point of each interval and evaluate the sign in each:

Interval

P(0) = 02 − 6 · 0 + 8 > 0

P(3) = 32 − 6 · 3 + 8 = 17 − 18 < 0

P(5) = 52 − 6 · 5 + 8 = 33 − 30 > 0

3. The solution is defined by the intervals (or the interval) that have the same sign as the polynomial.

Interval

S = (-∞, 2) Unión (4, ∞)


x2 + 2x +1 ≥ 0

x2 + 2x +1 = 0

Quadratic Inequality

(x + 1)2 ≥ 0

As a number squared is always positive the solution is Any Real Number.

    Solution
x2 + 2x +1 ≥ 0 (x + 1)2 ≥ 0 R
x2 + 2x +1 > 0 (x + 1)2 > 0 R-1
x2 + 2x +1 ≤ 0 (x + 1)2 ≤ 0 x = − 1
x2 + 2x +1 < 0 (x + 1)2 < 0 vacio

x2 + x +1 > 0

x2 + x +1 = 0

Quadratic Inequality Solution


When there are no real roots, give the polynomial any value if:

The sign obtained coincides with the inequality, the solution is R.

The obtained sign does not coincide with that of the inequality and thus, has no solution.

  Solution
x2 + x +1 ≥ 0 R
x2 + x +1 > 0 R
x2 + x +1 ≤ 0 vacio
x2 + x +1 < 0 vacio

Examples

1 7x2 + 21x − 28 < 0

x2 +3x − 4 < 0

x2 +3x − 4 = 0

Quadratic Inequality Operations

P(−6) = (−6)2 +3 · (−6)− 4 > 0

P(0) = 02 +3 · 0 − 4 < 0

P(3) = 32 +3 · 3 − 4 > 0

Quadratic Inequality Interval

(−4, 1)

2 −x2 + 4x − 7 < 0

x2 − 4x + 7 = 0

Quadratic Inequality Operations

P(0) = −02 + 4 ·0 − 7 < 0

S = R

3inecuación

Quadratic Inequality Operations

Interval

P(−3) = 4 · (−3)2 − 16 > 0

P(0) = 4 · 0 2 − 16 < 0

P(3) = 4 · 3 2 − 16 > 0

Interval

(-∞, −2] Unión [2, +∞)