Question Resuelva la inecuación cuadrática $x^2 - 5x - 14 \geq 0$ .

Studdy Solution

STEP 1

Assumptions1. The quadratic inequality is $x^{}-5x-14 \geq0$ . We are looking for the values of $x$ that satisfy this inequality.

STEP 2

First, we need to factor the quadratic expression. This can be done by finding two numbers that multiply to -14 and add to -5.

STEP 3

The two numbers that satisfy these conditions are -7 and2. So, we can rewrite the quadratic expression as follows $(x -7)(x +2) \geq0$

STEP 4

Now, we need to find the roots of the equation $(x -7)(x +2) =0$ . These are the values of $x$ that make the expression equal to zero.

STEP 5

Setting $x -7 =0$ , we find that $x =7$ .

STEP 6

Setting $x +2 =0$ , we find that $x = -2$ .

STEP 7

Now, we have three intervals to consider $(- \infty, -2)$ , $(-2,7)$ , and $(7, \infty)$ . We need to test the sign of the expression $(x -7)(x +2)$ in each of these intervals.

STEP 8

Choose a test point in the interval $(- \infty, -2)$ , say $x = -3$ . Substituting this into the expression, we get $(-3 -7)(-3 +2) = (-10)(-1) =10 >0$ . So, the expression is positive in this interval.

STEP 9

Choose a test point in the interval $(-2,7)$ , say $x =$ . Substituting this into the expression, we get $( -7)( +2) = (-7)(2) = -14 <$ . So, the expression is negative in this interval.

STEP 10

Choose a test point in the interval $(7, \infty)$ , say $x =8$ . Substituting this into the expression, we get $(8 -7)(8 +2) = ()(10) =10 >0$ . So, the expression is positive in this interval.

STEP 11

Since we want the expression to be greater than or equal to zero, the solution to the inequality is the union of the intervals where the expression is non-negative. This is $(- \infty, -] \cup [7, \infty)$ .
The solution to the inequality $x^{}-5x-14 \geq0$ is $x \in (- \infty, -] \cup [7, \infty)$ .

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