Problem 75 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the inequality and graph the solution on the real number line. Use a graphing utility to verify your solution graphically. $$4 x^{2}-4 x+1>0$$

Step-by-Step Solution

Answer

The inequality holds for all real numbers. Thus, the solution set of \(4x^2 - 4x + 1 > 0\) is \((-∞, +∞)\).

Step 1: Solve the Quadratic Equation

Set the inequality \(4x^2 - 4x + 1 > 0\) as an equation \(4x^2 - 4x + 1 = 0\) and solve for x. You can use the quadratic formula to solve this equation. The formula is \(x = [-b ± sqrt(b^2 - 4ac)]/(2a)\), where a = 4, b = -4 and c = 1. However, the discriminant (b^2 - 4ac) is less than 0. So there are no real roots.

Step 2: Check Inequality

The quadratic function \(4x^2 - 4x + 1\) does not touch or cross the x-axis since it doesn't have real roots. But since the quadratic coefficient is positive, which means that the parabola opens upwards, the function is always greater than zero for all real x.

Step 3: Graph the Function and Inequality

Draw a graph of the function \(4x^2 - 4x + 1\). It is a parabola that opens upwards and is above the x-axis for all x. Therefore, \(4x^2 - 4x + 1 > 0\) is true for all real numbers (extend the line to the left and right to represent all real numbers) You can use a graphing utility to confirm.

Key Concepts

Quadratic Formula

The quadratic formula is an essential tool for solving quadratic equations, which are polynomial equations of the second degree. Essentially, if you have a quadratic equation in the standard form of \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, you can solve for \(x\) using the quadratic formula: \(x = [-b \pm \sqrt{b^2 - 4ac}]/(2a)\).

Here, '\(\pm\)' signifies that you'll often get two solutions, which correspond to the two points where a parabola intersects the x-axis. The caveat is that this only applies when the solutions are real numbers. If the discriminant (\(b^2 - 4ac\)) is negative, there will be no real intercepts on the x-axis. In the case of the inequality \(4x^2 - 4x + 1 > 0\), the discriminant is less than zero, indicating that the quadratic equation has no real solutions and, therefore, the parabola does not cross the x-axis.

Graphing Quadratic Functions

Graphing quadratic functions is a visual way to understand the properties and behavior of quadratics. A quadratic function, written as \(f(x) = ax^2 + bx + c\), graphs into a parabola. Depending on the sign of \(a\), the parabola opens upwards (if \(a > 0\)) or downwards (if \(a < 0\)).

Vertex and Axis of Symmetry

Every parabola has a vertex, which is either the highest or lowest point on the graph, and an axis of symmetry, a vertical line that passes through the vertex. The vertex can be found using the formula \(x = -b/(2a)\), and this x-value is plugged back into the quadratic function to find the y-value of the vertex.

Intercepts and Direction

The point where the parabola intersects the x-axis is known as the x-intercept, and the y-intercept is where the function crosses the y-axis. The number of x-intercepts depends on the discriminant's value. In our inequality example, the parabola of the function \(4x^2 - 4x + 1\) opens upwards because \(a\) is positive, and has no x-intercepts as discussed previously.

Discriminant Analysis

Discriminant analysis isn't about a method for directly solving inequalities or equations but rather a way of determining the nature of the solutions we can expect from a quadratic equation. The discriminant, given by the expression \(b^2 - 4ac\), is a component of the quadratic formula and informs us about the number and type of solutions.

If the discriminant is positive, the quadratic equation has two distinct real roots. If it is zero, there is exactly one real root, meaning the parabola touches the x-axis at one point (the vertex). For a negative discriminant, as in our example with \(4x^2 - 4x + 1 > 0\), there are no real roots; the parabola does not intersect the x-axis at all, which affects how we interpret the solution set of the inequality.