Problem 24 - College Algebra and Calculus: An Applied Approach

Question

Use the Quadratic Formula to solve the quadratic equation. $$ 16 x^{2}-40 x+5=0 $$

Step-by-Step Solution

Answer

The solutions for the equation \(16x^{2}-40x+5 = 0\) are \(x_{1} = 2.5 - 0.625\sqrt{5}\) and \(x_{2} = 2.5 + 0.625\sqrt{5}\)

Step 1: Identify a, b, and c

In the equation \(16x^{2}-40x+5 = 0\), a, b, and c are coefficients. In this case, \(a = 16\), \(b = -40\), and \(c = 5\).

Step 2: Calculate the Discriminant

The next step is to calculate the Value of the Discriminant using the formula \(D = b^{2} - 4ac\). Substitute \(a = 16\), \(b = -40\), and \(c = 5\) into formula. Therefore, \(D = (-40)^{2} - 4 * 16 * 5 = 1600 - 320 = 1280\).

Step 3: Compute the values of x

Substitute a, b, and D into the Quadratic Formula \[x = \frac{-b \pm \sqrt{D}}{2a}\]. So, \[x_{1,2} = \frac{-(-40) \pm \sqrt{1280}}{2*16}\]Solving gives \[x_{1} = \frac{40 + \sqrt{1280}}{32} = 2.5 - 0.625\sqrt{5}\]\[x_{2} = \frac{40 - \sqrt{1280}}{32} = 2.5 + 0.625\sqrt{5}\]

Key Concepts

Quadratic Formula

The Quadratic Formula is a key tool for solving quadratic equations, which are polynomials of the second degree. When an equation is in the form ax² + bx + c = 0, the solution for x can be found using the formula: \[ x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \]
This formula provides the roots of the quadratic equation by determining the values of x where the parabola (the graph of the equation) crosses the x-axis.

It comprises of two parts, one with a '+' (plus) sign and another with a '−' (minus) sign. These correspond to the two possible solutions for x, often referred to as x₁ and x₂. Utilizing this formula, you can solve any quadratic equation, even if the equation cannot be factored easily or at all. The Quadratic Formula is especially useful when the coefficients of the equation make factoring complicated, as seen in the textbook example.

Discriminant in Quadratic Equations

The discriminant in a quadratic equation is a critical component that tells us about the nature of the roots without actually calculating them. Defined as D = b² − 4ac, the discriminant can reveal whether the roots are real or complex, and whether they are distinct or repeated.

If D is positive, there are two distinct real roots. A zero discriminant indicates a single repeated real root, and a negative discriminant suggests two complex roots. In the given exercise, the discriminant is positive (1280), confirming that there are two different real solutions for x. The discriminant doesn't just describe the type of roots, but also their quantity, playing a pivotal role in the quadratic formula itself as we take the square root of the discriminant when solving for the roots.

Quadratic Coefficients

In a quadratic equation represented by ax² + bx + c = 0, the numbers a, b, and c are known as the quadratic coefficients. They are essential since they determine the shape, position, and orientation of the parabola produced by the equation.

The coefficient a affects the width and the direction (upwards or downwards) of the parabola. If a is positive, the parabola opens upwards, and if it's negative, it opens downwards. The coefficient b influences the position of the parabola along the x-axis, while c represents the y-intercept or the point where the parabola crosses the y-axis. In our exercise, we have a = 16, b = -40, and c = 5, allowing us to determine the orientation and width of the parabola, as well as to eventually find the solutions to the equation using the quadratic formula.