Stepwise Quadratic Inequalities Solving Guide

Quadratic inequalities often appear in math and engineering problems. Solving them correctly requires a clear method. This stepwise solving quadratic inequalities guide will help you find solutions easily and accurately. Following steps in order simplifies the process and reduces mistakes. Let’s explore the guide step by step.

Step 1: Rewrite the Inequality in Standard Form

Always start by rewriting the quadratic inequality so one side equals zero. For example, if you have x2>4x−5x^2 > 4x – 5x2>4x−5, subtract all terms on the right side to get x2−4x+5>0x^2 – 4x + 5 > 0x2−4x+5>0. Having zero on one side makes it easier to find roots and test intervals.

Step 2: Solve the Corresponding Quadratic Equation

Next, solve the quadratic equation formed by replacing the inequality sign with an equals sign. In the example above, solve x2−4x+5=0x^2 – 4x + 5 = 0x2−4x+5=0. Use factoring if possible, or apply the quadratic formula:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}x=2a−b±b2−4ac​​

Find the roots x1x_1x1​ and x2x_2x2​. These roots split the number line into intervals.

Step 3: Determine the Parabola’s Direction

Look at the coefficient of x2x^2x2, called $a$. If $a>0$, the parabola opens upward like a “U”. If $a<0$, it opens downward like an upside-down “U”. This direction tells you whether the quadratic is positive or negative between and outside the roots.

Step 4: Divide the Number Line Into Intervals

Use the roots found to split the number line into sections. For roots x1x_1x1​ and x2x_2x2​, you get three intervals: (−∞,x1)(-\infty, x_1)(−∞,x1​), (x1,x2)(x_1, x_2)(x1​,x2​), and (x2,∞)(x_2, \infty)(x2​,∞). Each interval will be tested in the next step.

Step 5: Select Test Points in Each Interval

Choose one number inside each interval. For example, if the intervals are $(-\infty ,2)$, $(2,5)$, and $(5,\infty )$, pick $x=0$, $x=3$, and $x=6$. Substituting these points into the original inequality helps you find if the inequality holds in that interval.

Step 6: Substitute Test Points Into the Inequality

Plug each test point into the original quadratic expression (not the equation). Check if the inequality is true or false for that point. For example, if the inequality is x2−4x+5>0x^2 – 4x + 5 > 0x2−4x+5>0, substitute $x=0$ to get $0-0+5=5>0$, which is true. Do this for all intervals.

Step 7: Write the Solution Using Interval Notation

Based on test results, write all intervals where the inequality is true. Use brackets $[]$ if the inequality includes equality ($\ge$ or $\le$) and parentheses $()$ if strict ($>$ or $<$). For example, if true for $(-\infty ,2)$ and $(5,\infty )$, write $(-\infty ,2)\cup (5,\infty )$.

Step 8: Check for Special Cases

If the quadratic equation has no real roots (discriminant < 0), the parabola never touches the x-axis. Then, the quadratic is always positive or always negative. Depending on the inequality, the solution might be all real numbers or no solution. Remember to consider this case.

Step 9: Verify Your Solution

Finally, verify by graphing or double-checking your intervals. A quick sketch of the parabola confirms if your solution matches the parts above or below the x-axis. Verification prevents mistakes and increases confidence.

Example: Solve x2−5x+6≤0x^2 – 5x + 6 \leq 0x2−5x+6≤0 Stepwise

Step 1: Write in standard form — already x2−5x+6≤0x^2 – 5x + 6 \leq 0x2−5x+6≤0.

Step 2: Solve the equation x2−5x+6=0x^2 – 5x + 6 = 0x2−5x+6=0. Factor as $(x-2)(x-3)=0$. Roots: 2 and 3.

Conclusion

Following this stepwise quadratic inequalities solving guide helps you find solutions clearly and accurately. Always rewrite in standard form, solve the related equation, test intervals, and express the solution using interval notation. Practice these steps regularly to become confident in solving quadratic inequalities.