Problem 28 - Contemporary Precalculus

Question

Find the number of real solutions of the equation by computing the discriminant. $$4 x^{2}-4 x-3=0$$

Step-by-Step Solution

Answer

Answer: The quadratic equation has two distinct real solutions.

Step 1: Identify the coefficients

The given equation is $$4x^2 - 4x - 3 = 0$$. We can see that the coefficients are \(a = 4\), \(b = -4\), and \(c = -3\).

Step 2: Calculate the discriminant

Now, we will compute the discriminant \(\Delta\) using the formula \(\Delta = b^2 - 4ac\). In this case, \(\Delta = (-4)^2 - 4(4)(-3)\).

Step 3: Evaluate the expression

Evaluating the expression, we find that $$ \Delta = 16 + 48 = 64 $$

Step 4: Determine the number of real solutions

Since the discriminant \(\Delta = 64\) is positive, the given quadratic equation has two distinct real solutions.

Key Concepts

Quadratic Equations

A quadratic equation is a second-degree polynomial, typically expressed in the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(x\) represents an unknown variable. The number \(a\) is known as the leading coefficient and must not be zero, otherwise, the equation would be linear, not quadratic. In the context of our exercise, \(4x^2 - 4x - 3 = 0\) is a classic example of a quadratic equation.
In finding solutions to a quadratic equation, we often look to the roots, which can be real or complex numbers where the equation equals zero. These roots can be calculated using various methods such as factoring, completing the square, using the Quadratic Formula, or graphing. The most efficient method often depends on the form of the quadratic equation given.

Real Solutions of Equations

The nature of the solutions to quadratic equations is heavily dependent on the discriminant, denoted \(\Delta\). Real solutions of an equation refer to the x-values that satisfy the equation, which, when plotted on a graph, correspond to the points where the parabola (the graph of a quadratic equation) intersects the x-axis.
A quadratic equation can have two real and distinct solutions, one real and repeated solution, or no real solutions at all. If \(\Delta > 0\), the equation has two distinct real solutions; if \(\Delta = 0\), there is exactly one real solution; and if \(\Delta < 0\), the equation has no real solutions, but rather two complex solutions. In our sample exercise, the discriminant value of 64, which is positive, indicates that there are two distinct real solutions for the quadratic equation.

Calculating Discriminants

To calculate the discriminant of a quadratic equation \(ax^2 + bx + c = 0\), you use the formula \(\Delta = b^2 - 4ac\). This calculation is a crucial step in determining the nature and number of solutions to the equation.

• When \(\Delta > 0\), two distinct real roots exist, and the parabola crosses the x-axis at two points.
• When \(\Delta = 0\), there is one real root (also called a repeated or double root), where the parabola just touches the x-axis.
• When \(\Delta < 0\), no real roots exist, and the parabola does not intersect the x-axis; instead, there are two complex roots.

In the exercise provided, the discriminant is calculated as \(\Delta = (-4)^2 - 4(4)(-3)\), which simplifies to \(\Delta = 16 + 48 = 64\). Since the discriminant is positive, it confirms the presence of two distinct real solutions for the equation \(4x^2 - 4x - 3 = 0\).