Question

Write a quadratic equation in standard form that has roots equidistant from 10 on the number line.

Short answer

The quadratic equation in standard form that has roots equidistant from 10 on the number line is \(y = ax^2 - 20ax + 100a - ad^2\).

Step-by-step solution

Determine the Roots

The roots are equidistant from 10 on the number line. Let \(d\) be the common distance. Therefore, one root is \(10 + d\) and the other root is \(10 - d\). So, roots are \(10 - d\) and \(10 + d\).

Construct the Quadratic Equation from the Roots

Recall that a quadratic equation in the form of \(y = a(x - p)(x - q)\) will have roots at \(p\) and \(q\). Therefore, we can construct a quadratic equation using our roots: \(y = a(x - (10 - d))(x - (10 + d))\). which simplifies to \(y = a(x - 10 + d)(x - 10 -d)\). This is the quadratic equation with the roots equidistant from 10.

Write quadratic equation in standard form

We can put our equation in standard form by expanding the expression: \(y = a[(x - 10)^2 - d^2]\). This expands to \(y = ax^2 - 20ax + 100a - ad^2\). This is the quadratic equation in standard form that has roots equidistant from 10 on the number line.

Key concepts

Roots of a Quadratic Equation

The roots of a quadratic equation, also known as solutions or x-intercepts, are the values of x that make the equation equal zero. These roots can be real or complex numbers. When plotted on a graph, they represent the points where the curve of a quadratic function intersects the x-axis.

In the context of the exercise, the quadratic equation is designed to have roots that are equidistant from a specific point on the number line, which is the number 10 in this case. To find these roots, one can simply add and subtract the same distance, denoted as 'd', from the point of reference. This leads to roots of the form \(10 + d\) and \(10 - d\), illustrating the symmetric nature of these roots with respect to the point on the number line. Understanding roots in this way allows us to generate quadratic equations with designated properties and to visualize the position of roots in relation to the entire set of real numbers.

Constructing Quadratic Equations

Formulating a quadratic equation based on given roots involves using the factored form of the equation, which is \(y = a(x - p)(x - q)\), where 'p' and 'q' are the roots. The constant 'a' can be any non-zero value and determines the vertical stretch or compression of the parabola.

When constructing a quadratic equation from roots, we make use of the fact that the product of two binomials will yield a quadratic expression. Applying this method to our exercise with roots \(10 - d\) and \(10 + d\), we set up our function \(y = a(x - (10 - d))(x - (10 + d))\), further simplifying it to match the standard form which is \(ax^2 + bx + c\). The standard form will help us understand the attributes of the parabola such as the direction of opening, vertex, and axis of symmetry. This process of constructing equations from roots is fundamental in algebra and provides a systematic approach to creating equations that model various real-world phenomena.

Number Line Concepts

A number line is a crucial illustrated representation used to order numbers and to provide a visual understanding of numerical relationships. It helps in comprehending the concept of distance between numbers, which is highly relevant in our exercise.

On a number line, points that are equidistant from a central point mirror each other across that central point. This symmetry is intrinsic to the exercise scenario of roots equidistant from the number 10. The roots, located at the same distance but on opposite sides of 10, are essentially mirrored on the number line.

Coupled with an understanding of how quadratic functions behave, the number line aids us in anticipating the placement of the x-intercepts (roots) of a quadratic equation and their relationship with each other. In practical terms, this conceptual framework assists students in both constructing equations and in solving them, substantiating their algebraic knowledge with a solid visual backing.