Share
Solving Quadratic Inequalities Step-by-Step
Published on April 30, 2026
Algebra
Learn how to solve various quadratic inequalities with detailed step-by-step explanations, including finding roots and testing intervals.
Question
resuelve paso a paso la inecuaciones de segundo grado en la imagen adjunta
Answer
Here are the step-by-step solutions for each inequality:
a)
-
Find the roots of the quadratic equation
.
We can factor this as.
The roots areand . -
Determine the intervals.
The roots divide the number line into three intervals:, , and . -
Test a value from each interval.
- For
, let’s pick : . - For
, let’s pick : . - For
, let’s pick : .
- For
-
State the solution.
Since the inequality is $ > 0$, we want the intervals where the expression is positive.
The solution is.
b)
-
Find the roots of
.
Factoring gives.
The roots areand . -
Determine the intervals.
The intervals are, , and . -
Test a value from each interval.
- For
, let : . - For
, let : . - For
, let : .
- For
-
State the solution.
The inequality is $ > 0$.
The solution is.
c)
-
Find the roots of
.
Factoring gives.
The roots areand . -
Determine the intervals.
The intervals are, , and . -
Test a value from each interval.
- For
, let : . - For
, let : . - For
, let : .
- For
-
State the solution.
The inequality is $ \ge 0$. We include the roots because of the “equal to” part.
The solution is.
d)
-
Find the roots of
.
Factoring gives.
The roots areand . -
Determine the intervals.
The intervals are, , and . -
Test a value from each interval.
- For
, let : . - For
, let : . - For
, let : .
- For
-
State the solution.
The inequality is $ < 0$.
The solution is.
e)
-
Find the roots of
.
Factoring gives.
The roots areand . -
Determine the intervals.
The intervals are, , and . -
Test a value from each interval.
- For
, let : . - For
, let : . - For
, let : .
- For
-
State the solution.
The inequality is $ < 0$.
The solution is.
f)
-
Find the roots of
.
Factoring gives.
The roots areand . -
Determine the intervals.
The intervals are, , and . -
Test a value from each interval.
- For
, let : . - For
, let : . - For
, let : .
- For
-
State the solution.
The inequality is $ \le 0$. We include the roots.
The solution is.
g)
-
Find the roots of
.
Factoring gives.
The roots areand . -
Determine the intervals.
The intervals are, , and . -
Test a value from each interval.
- For
, let : . - For
, let : . - For
, let : .
- For
-
State the solution.
The inequality is $ \le 0$. We include the roots.
The solution is.
h)
-
Find the roots of
.
Factoring gives.
The roots areand . -
Determine the intervals.
The intervals are, , and . -
Test a value from each interval.
- For
, let : . - For
, let : . - For
, let : .
- For
-
State the solution.
The inequality is $ \ge 0$. We include the roots.
The solution is.
i)
-
Find the roots of
.
Multiply by -1 to make the leading coefficient positive:.
Factoring gives.
The roots areand . -
Determine the intervals.
The intervals are, , and . -
Test a value from each interval.
- For
, let : . - For
, let : . - For
, let : .
- For
-
State the solution.
The inequality is $ \ge 0$. We include the roots.
The solution is.
j)
-
Find the roots of
.
Multiply by -1:.
Factoring gives.
The roots areand . -
Determine the intervals.
The intervals are, , and . -
Test a value from each interval.
- For
, let : . - For
, let : . - For
, let : .
- For
-
State the solution.
The inequality is $ > 0$.
The solution is.
k)
-
Find the roots of
.
Multiply by -1:.
Factoring gives.
The roots areand . -
Determine the intervals.
The intervals are, , and . -
Test a value from each interval.
- For
, let : . - For
, let : . - For
, let : .
- For
-
State the solution.
The inequality is $ < 0$.
The solution is.
l)
-
Find the roots of
.
Multiply by -1:.
Factoring gives.
The roots areand . -
Determine the intervals.
The intervals are, , and . -
Test a value from each interval.
- For
, let : . - For
, let : . - For
, let : .
- For
-
State the solution.
The inequality is $ \ge 0$. We include the roots.
The solution is.
m)
-
Find the roots of
.
Multiply by -1:.
Factoring gives.
The roots areand . -
Determine the intervals.
The intervals are, , and . -
Test a value from each interval.
- For
, let : . - For
, let : . - For
, let : .
- For
-
State the solution.
The inequality is $ \le 0$. We include the roots.
The solution is.
n)
-
Find the roots of
.
Multiply by -1:.
Factoring gives.
The roots areand . -
Determine the intervals.
The intervals are, , and . -
Test a value from each interval.
- For
, let : . - For
, let : . - For
, let : .
- For
-
State the solution.
The inequality is $ < 0$.
The solution is.