Problem 29 - Algebra 1: Concepts and Skills

Question

Determine whether the equation has two solutions, one solution, or no real solution. \(4 x^{2}-5 x+1=0\)

Step-by-Step Solution

Answer

The quadratic equation has two real and distinct solutions.

Step 1: Identify a, b, and c in the equation

In the given quadratic equation \(4 x^{2}-5x+1 = 0\), the coefficients are \(a = 4\), \(b = -5\), and \(c = 1\).

Step 2: Calculate the Discriminant

Determine the value of the discriminant using the formula \(D=b^{2}-4ac\). Plugging in the coefficients into the formula gives us \(D = (-5)^{2}-4*(4)*(1)\). This simplifies to \(D = 25 - 16 = 9\).

Step 3: Determine the type of solutions

Since \(D > 0\), the conclusion is that the given equation has two real and distinct solutions.

Key Concepts

Discriminant of a Quadratic Equation

The discriminant of a quadratic equation is a part of the quadratic formula that provides crucial information about the nature of the solutions to the equation. The general form of a quadratic equation is \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are coefficients, with \(a \eq 0\).

The discriminant, usually represented by the symbol \(D\), is calculated as \((D = b^2 - 4ac)\). The value of the discriminant tells us the number and type of solutions:

• If \(D > 0\), there are two real and distinct solutions.
• If \(D = 0\), there is exactly one real solution, also known as a repeated or double root.
• If \(D < 0\), there are no real solutions; instead, there are two complex solutions.

In our exercise, after identifying the coefficients as \(a = 4\), \(b = -5\), and \(c = 1\), and calculating the discriminant as \(D = 9\), we see that \(D > 0\). This indicates that our quadratic equation has two real and distinct solutions.

Quadratic Formula

The quadratic formula is a powerful tool used to find the solutions of a quadratic equation. The formula is derived from completing the square method and is given by: \[-\frac{b}{2a} \pm \sqrt{\frac{b^2-4ac}{4a^2}}\] which simplifies to the familiar \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\].

The quadratic formula includes the discriminant \(D = b^2 - 4ac\), and the sign of the square root \(\sqrt{D}\) determines the number of solutions. The term \(\pm\) indicates that there are two solutions for \(x\), corresponding to the two possible values—adding or subtracting the square root of the discriminant.

To apply this formula to our example, we would plug in our coefficients:

Solution Using Quadratic Formula

\[x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(4)(1)}}{2(4)}\]
This would provide us with the precise values of \(x\) for our two real solutions.

Real Solutions of Quadratic Equations

Real solutions of a quadratic equation are the x-values that satisfy the equation, where the curve of the quadratic function intersects the x-axis. As we have seen, the discriminant \(D\) helps us to determine the nature of these solutions.

If the discriminant is positive, as in our example with \(D = 9\), there are two different points where the parabola crosses the x-axis, which means there are two distinct real solutions. If the discriminant is zero, the parabola touches the x-axis at a single point, representing a repeated real solution. When the discriminant is negative, the parabola does not intersect the x-axis at all, indicating that there are no real solutions.

To find the exact real solutions, when available, we use the quadratic formula. In practical cases, solutions to quadratic equations allow us to determine key points in various scientific and engineering contexts, such as the apex of a projectile's trajectory, or the points at which a product's profit function maximizes.