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Gather Your Inputs
First, identify the coefficients $a$, $b$, and $c$ in the quadratic inequality. For example, in the inequality $x^2 + 4x + 4 \geq 0$, we have $a = 1$, $b = 4$, and $c = 4$.
Find the Critical Points
Next, find the critical points by solving the quadratic equation $ax^2 + bx + c = 0$. Use the quadratic formula: $x = rac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. For our example, $x = rac{-4 \pm \sqrt{4^2 - 4(1)(4)}}{2(1)} = rac{-4 \pm \sqrt{16 - 16}}{2} = rac{-4 \pm \sqrt{0}}{2} = -2$.
Analyze the Parabola
Analyze the parabola by determining its opening direction. If $a > 0$, the parabola opens upwards, and if $a < 0$, it opens downwards. In our example, $a = 1 > 0$, so the parabola opens upwards.
Determine the Solution Set
Determine the solution set by considering the intervals where the quadratic expression is positive or negative. Since the parabola opens upwards and the critical point is $x = -2$, the solution set for the inequality $x^2 + 4x + 4 \geq 0$ is $x \leq -2$ or $x \geq -2$. However, since the inequality is $\geq 0$, the solution set is $x \geq -2$ or $x \leq -2$, which simplifies to $x = -2$ because the quadratic expression is always positive except at $x = -2$ where it equals zero.
Express the Solution Set in Interval Notation
Express the solution set in interval notation. For our example, the solution set is $x = -2$, which can be written as $[-2, -2]$ or simply $x = -2$.
Common Mistakes to Avoid and Using a Calculator
Common mistakes to avoid include forgetting to consider the opening direction of the parabola and neglecting to check for extraneous solutions. While calculators can be convenient for solving quadratic inequalities, it's essential to understand the underlying concepts and be able to solve them manually. Use a calculator to check your solutions or when dealing with complex inequalities.
Introduction to Quadratic Inequalities
Quadratic inequalities are expressions of the form $ax^2 + bx + c \geq 0$ or $ax^2 + bx + c \leq 0$, where $a$, $b$, and $c$ are real numbers. To solve these inequalities, we need to find the values of $x$ that make the expression true.
Prerequisites
Before we dive into the steps, make sure you have a basic understanding of quadratic equations and their graphs. The formula for a quadratic equation is $ax^2 + bx + c = 0$, and the solutions can be found using the quadratic formula: $x = rac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
Understanding the Formula
The formula for solving quadratic inequalities is based on the sign of the quadratic expression $ax^2 + bx + c$. We need to find the intervals where the expression is positive or negative.
Step-by-Step Solution
To solve a quadratic inequality, follow these steps: