Solve Quadratic Inequalities Graphically

This is a tutorial on how to solve quadratic inequalities graphically. The quadratic inequalities explored are of the type \[a x^2 + b x + c \lt 0\] and \[a x^2 + b x + c \gt 0\]

Review

An app plots the graph of \(y = a x^2 + b x + c\) and displays part of the graph that is below the \(x\)-axis \(( y \lt 0 )\) in blue and part of the graph that is above the \(x\)-axis \(( y \gt 0 )\) in red. To solve a quadratic inequality, you just read the interval corresponding to \(y \lt 0\) or \(y \gt 0\) depending on the inequality to solve.

Interactive Tutorials

Quadratic Function Plotter

Enter values for coefficients \(a\), \(b\), and \(c\) to see how they affect the parabola shape and position.
Positive y values
Negative y values
X-intercepts
Vertex
$$y = ax^2 + bx + c$$
Current equation: \(y = 1.0 x^2 + 2x - 3\)

Examples & Step-by-Step Solutions

Example 1: Quadratic Inequality \(-x^2 + 3x + 4 \lt 0\)

Problem: Solve graphically and analytically the quadratic inequality

\[- x^2 + 3x + 4 \lt 0\]

  1. Graphical Solution: Use the app above to enter coefficients \(a = -1\), \(b = 3\), and \(c = 4\) and graph the equation \(y = -x^2 + 3x + 4\). The solution set to the inequality \(-x^2 + 3x + 4 \lt 0\) corresponds to the x-coordinates of the points on the graph for which \(y \lt 0\) (displayed in BLUE). We have two intervals for whose union is written in interval form as: \[(-\infty, -1) \cup (4, +\infty)\]
  2. Analytical Solution:
    • Factor the left-hand term of the given inequality: \[-x^2 + 3x + 4 = (x + 1)(-x + 4)\]
    • To solve the given inequality, we study the sign of the expression: \[(x + 1)(-x + 4)\] This expression has zeros at \(x = -1\) and \(x = 4\). These values divide the number line into three intervals:
      • \((-\infty, -1)\)
      • \((-1, 4)\)
      • \((4, +\infty)\)
    • We analyze the sign of the expression \(-x^2 + 3x + 4\) on each interval by choosing test points:
      1. Interval \((-\infty, -1)\): Let \(x = -2\) \[-(-2)^2 + 3(-2) + 4 = -4 - 6 + 4 = -6\] The result is negative, so this interval is part of the solution set.
      2. Interval \((-1, 4)\): Let \(x = 0\) \[-(0)^2 + 3(0) + 4 = 4\] The result is positive, so this interval is not part of the solution set.
      3. Interval \((4, +\infty)\): Let \(x = 5\) \[-(5)^2 + 3(5) + 4 = -25 + 15 + 4 = -6\] The result is negative, so this interval is part of the solution set.
    • Final Answer: The solution set of the inequality is the union of the two intervals over which the expression is negative: \[(-\infty, -1) \cup (4, +\infty)\]

    Note: Both the graphical and analytical methods give the same answer.

Example 2: Quadratic Inequality \(-x^2 + 4x - 5 \gt 0\)

Problem: Solve graphically and analytically the quadratic inequality

\[-x^2 + 4x - 5 \gt 0\]

  1. Graphical Solution: Use the app to set coefficients \(a = -1\), \(b = 4\), and \(c = -5\) and graph the equation \(y = -x^2 + 4x - 5\). This inequality has no solutions since the entire graph lies below the \(x\)-axis, meaning \(-x^2 + 4x - 5 \lt 0\) for all real values of \(x\).
  2. Analytical Solution:
    • The expression \(-x^2 + 4x - 5\) cannot be factored over the real numbers because its discriminant \[\Delta = 4^2 - 4(-1)(-5) = 16 - 20 = -4\] is negative. Therefore, it has no real zeros and its sign does not change across the real number line.
    • To determine its sign, we evaluate the expression at a single value of \(x\). Let's choose \(x = 0\): \[-(0)^2 + 4(0) - 5 = -5\] Since the result is negative, the expression is negative for all real values of \(x\).
    • Final Answer: The expression \(-x^2 + 4x - 5\) is always negative. Therefore, the inequality \(-x^2 + 4x - 5 > 0\) has no solution.

Exercises

Solve each quadratic inequality both graphically (using the app) and analytically:

  1. \[-x^2 - 4x \lt -5\]
  2. \[x^2 - 2x + 8 \geq 0\]
  3. \[x^2 - 3x \leq 0\]

Solutions to the Above Exercises

  1. \[(-\infty, -5) \cup (1, +\infty)\]
  2. \[(-\infty, +\infty)\]
  3. \[[0, 3]\]

More References and Links

Explore additional equations, inequalities, and math tutorials:

Solve Equations, Systems of Equations and Inequalities | Math Problems, Questions and Tests | Solve Inequalities in One Variable | Home Page