Question

Write a quadratic equation with the given solutions. 427 and 0

Short answer

The quadratic equation with roots 427 and 0 is \(x^2 - 427x = 0\)

Step-by-step solution

Identify the Roots

The roots given for the quadratic equation are 427 and 0.

Apply Sum and Product of Roots in Quadratic Formula

In the quadratic formula \(x^2 - (sum \, of \, roots)x + product \, of \, roots = 0\), the sum of roots is equal to 427+0 = 427 while the product of roots is 427*0 = 0.

Substitute the Values to Get the Equation

Now substituting the sum and product of roots values in equation, we get \(x^2 - 427x + 0 = 0\). On simplification, it becomes \(x^2 - 427x = 0\)

Key concepts

Sum and Product of Roots in Quadratic Equations

Understanding the relationship between the roots of a quadratic equation and the equation itself is crucial for solving quadratic problems. A quadratic equation generally takes the form of \(ax^2 + bx + c = 0\text{, where }\ a eq 0\). The fascinating aspect of quadratic equations is that the sum and product of its roots can be directly related to the coefficients of \(x\).

For any quadratic equation, the sum of its roots, denoted as \(\alpha + \beta\), is equal to the negation of the coefficient \(b\) divided by the coefficient \(a\). On the other hand, the product of its roots, \(\alpha\beta\), is equal to the constant term \(c\) divided by coefficient \(a\). Therefore, knowing the roots allows us to write the equation in the form \(x^2 - (\alpha + \beta)x + \alpha\beta = 0\). In the textbook exercise given, the sum of the roots, 427 + 0, guides us to the coefficient \(b\), while their product, 427\times0, corresponds to the constant term \(c\), resulting in a concise equation.

Roots of Quadratic Equations

The roots of a quadratic equation are the values of \(x\) that make the equation true. These roots can either be real or complex and are key to understanding the graph and characteristics of the equation. A quadratic equation may have two distinct roots, one repeated root, or no real roots at all, varying with the discriminant \(\Delta = b^2 - 4ac\).

The provided exercise specifies roots of 427 and 0, which are unique and very much real. Having these roots allows us to visualize the crossing points of the graph of the equation on the \(x\)-axis. The knowledge of these intersections is not only important for drawing the graph but also for completing various algebraic tasks that require factoring or applying the quadratic formula. In this scenario, we observe that one of the roots being zero simplifies the equation greatly, giving us a linear term in the polynomial.

The Quadratic Formula

The quadratic formula is derived from completing the square in the general quadratic equation \(ax^2 + bx + c = 0\) and provides a direct method to find the roots of any quadratic equation. The formula is given by \[x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\].

In practice, we use the quadratic formula when factoring is not feasible or to quickly determine the roots without factoring. The discriminant, the part of the formula under the square root \(\sqrt{b^2-4ac}\), dictates the nature and quantity of the roots.

For the exercise at hand, the quadratic formula can be used as a check. However, since one of the roots is already known to be zero, it simplifies to the linear form without the need for using the full quadratic formula. This demonstrates the power of understanding different methods to approach and solve quadratic equations.