Question

Use the Quadratic Formula to write a quadratic equation that has the given solutions.\(x=\frac{15 \pm \sqrt{-215}}{22}\)

Short answer

The quadratic equation that has the given solutions is \(22x^2 - 15x - \frac{215}{88} = 0\)

Step-by-step solution

Identify the coefficients from the solutions

First, observe the solutions provided, which is \(x = \frac{15 \pm \sqrt{-215}}{22}\). Notice that this is in the form of the quadratic formula, which is \(-\frac{b \pm \sqrt{b^2 - 4ac}}{2a}\). Map the coefficients a, b and c. In this case, \(a = 22\), \(b = -15\), and \(4ac = -215\)

Calculate the value of c

Next, calculate the value of c. Using the equation \(4ac = -215\), substitute a into the equation to get \(4*22*c = -215\). Solve for c to get \(c = \frac{-215}{88}\) or \(c = -\frac{215}{88}\)

Write the quadratic equation

Lastly, substitute a, b, and c into the general form of a quadratic equation \(ax^2 + bx + c = 0\) to get the quadratic equation. Plug in the obtained coefficients to form the equation: \(22x^2 - 15x - \frac{215}{88} = 0\).

Key concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation of degree two. It is typically written in the form \(ax^2 + bx + c = 0\), where:

• \(a\), \(b\), and \(c\) are known as the coefficients of the quadratic equation.
• \(a\) is not equal to zero; otherwise, the equation becomes linear.

The primary characteristic of a quadratic equation is its parabolic graph, which can open upwards or downwards depending on the sign of the leading coefficient \(a\). Quadratic equations are fundamental in algebra since they define curves and can be used to model various real-world scenarios like projectile motions and area problems.
To solve a quadratic equation, one might factor it, complete the square, or, as in our exercise, use the Quadratic Formula. This formula offers a method for finding solutions (or roots) of the quadratic equation, expressed as:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula grants solutions both for real and complex solutions, making it a comprehensive tool in solving quadratics.

Coefficients in Quadratics

Coefficients are critical components in quadratic equations since they determine the shape and position of the parabola on a graph. In a quadratic equation \(ax^2 + bx + c = 0\), each coefficient plays a distinct role:

• \(a\) controls the width and direction of the parabola.
• \(b\) influences the vertex's position along the x-axis.
• \(c\) decides where the parabola intersects the y-axis.

In the context of the quadratic formula, these coefficients directly impact the calculus of the solutions. In our exercise, identifying the coefficients from the given solution required matching the format \(-\frac{b \pm \sqrt{b^2 - 4ac}}{2a}\) and carefully rearranging the given algebraic fractions.
From learning how to determine these coefficients, students can gain a deeper understanding of how each element in the equation influences the overarching quadratic graph. This knowledge is essential when conceptualizing the nature of solutions, whether they are real or involve complex numbers.

Complex Solutions in Quadratics

Complex solutions arise in quadratic equations when the discriminant, \(b^2 - 4ac\), is negative. This results in the square root of a negative number, leading to what we call complex or imaginary numbers. A complex number is expressed as \(a + bi\), where:

• \(a\) is the real part.
• \(bi\) is the imaginary part, with \(i^2 = -1\).

In the exercise, the term \(\sqrt{-215}\) implies complex solutions since the radicand is negative. This indicates that the parabola doesn't intersect the x-axis at any real point, but instead at complex points in the complex plane. This mathematical concept is critical in various applications, including electrical engineering and quantum physics, where real numbers alone are insufficient. Recognizing and working with complex solutions not only extends algebra skills but also enriches mathematical comprehension, particularly in higher-level studies. It showcases the beauty and power of extending mathematics beyond traditional real-number limitations.