Solve each quadratic inequality. Give the solution set in interval notation. 4x²+3x+1≤0

Verified step by step guidance

1. Start by identifying the quadratic inequality: \(4x^{2} + 3x + 1 \leq 0\).
2. Find the roots of the corresponding quadratic equation \$4x^{2} + 3x + 1 = 0\( by using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\), where \)a=4\(, \)b=3\(, and \)c=1$.
3. Calculate the discriminant \(\Delta = b^{2} - 4ac = 3^{2} - 4 \times 4 \times 1\) to determine the nature of the roots.
4. Use the roots found to divide the number line into intervals. Test a value from each interval in the inequality \(4x^{2} + 3x + 1 \leq 0\) to determine where the inequality holds true.
5. Express the solution set as an interval or union of intervals based on where the inequality is satisfied, including endpoints if the inequality is non-strict (\(\leq\)).

Key Concepts

Quadratic Inequalities

A quadratic inequality involves a quadratic expression set less than, greater than, or equal to zero. Solving it requires finding the values of the variable that make the inequality true, often by analyzing the sign of the quadratic expression over different intervals.

Factoring and Quadratic Formula

To solve quadratic inequalities, you first find the roots of the corresponding quadratic equation by factoring or using the quadratic formula. These roots divide the number line into intervals where the quadratic expression may change sign.

Interval Notation and Sign Analysis

After finding the roots, test values from each interval to determine where the inequality holds. The solution set is then expressed in interval notation, which concisely represents all values satisfying the inequality.