Question

$$ 4 x^{2}-4 x y+y^{2}+7 x-5 y+8=0 $$

Short answer

The simplified expression of the given equation is \((2x-y)^{2} = (0.5y)^{2}-7x+5y-8\).

Step-by-step solution

Rearrange the terms

First of all, rearrange the terms according to the power of variables. The coefficients 4, -4 and 1 respectively multiply the terms \(x^{2}\), \(xy\) and \(y^{2}\). The coefficients 7 and -5 multiply single variable terms x and y respectively. Consider the constant as 8. Now regroup them as \(4x^{2}-4xy+y^{2}\) + \(7x-5y\)+8=0.

Completing the Square

The aim is to get a perfect square trinomial within the first three terms. A perfect square trinomial is a polynomial of the form \(a^{2} - 2ab + b^{2}\). Notice that if we add \((0.5y)^{2}\) lo the three terms, they will form a perfect square trinomial \((2x-y)^{2}\). However, to keep the equation balanced, we also need to subtract the same value. Thus, the equation now becomes \((2x-y)^{2} + 7x-5y+8 - (0.5y)^{2} = 0\).

Simplified Form

Rewrite the equation as \((2x-y)^{2} - (0.5y)^{2}+7x-5y+8 =0\). Now rearrange once more to get the final simplified equation: \((2x-y)^{2} = (0.5y)^{2}-7x+5y-8\).

Key concepts

Perfect Square Trinomial

A perfect square trinomial is a specific type of quadratic expression that can be written as the square of a binomial. It takes the form \(a^{2} + 2ab + b^{2}\) or \(a^{2} - 2ab + b^{2}\), where \(a\) and \(b\) are real numbers or algebraic expressions.

In the context of the given exercise, we aim to transform the portion of the quadratic \(4x^{2}-4xy+y^{2}\) into a perfect square trinomial. The equation hints at a hidden structure reminiscent of \(a^{2} - 2ab + b^{2}\), where \(a\) is doubled and \(b\) is derived from the coefficient of \(xy\). This insight leads to recognizing that this portion of the expression can indeed be rewritten as a perfect square: \(2x - y)^{2}\). This relies on the understanding that \(2x\) and \(y\) play the roles of \(a\) and \(b\), and when squared, yield the original terms of the quadratic component.

Quadratic Equation

A quadratic equation is a second-degree polynomial equation that can be expressed in the general form \(ax^{2}+bx+c=0\), where \(a\), \(b\), and \(c\) are coefficients, with \(a\) not equal to zero.

In our exercise, the initial equation \(4x^{2}-4xy+y^{2}+7x-5y+8=0\) is a quadratic equation in two variables, \(x\) and \(y\). Quadratic equations are pivotal in algebra because they pop up across various areas of mathematics and practical applications, ranging from physics to finance. By rearranging and solving these equations, we can find the values of the variables that satisfy the equation.

Algebraic Manipulation

Algebraic manipulation refers to the process of rearranging and simplifying algebraic expressions using the laws of arithmetic and algebra. The goal is to make an expression easier to understand or to solve an equation for an unknown variable.

During the exercise, we engaged in algebraic manipulation when we added \(0.5y)^{2}\) and then subtracted it to complete the square without changing the value of the original expression. It's a strategic move to facilitate the formation of a perfect square trinomial, thereby simplifying the equation and paving the way to a solution.

Polynomial Rearrangement

Polynomial rearrangement is a method used in algebra to reorganize the terms of a polynomial to either simplify the expression or prepare it for a specific method of solving, like completing the square or factoring.

In this exercise, we rearranged the polynomial by reordering its terms and regrouping them to highlight the structure necessary for making a perfect square. The equation rearrangement was \(4x^{2}-4xy+y^{2}\) and \(7x-5y\) as separate entities, setting up the subsequent step of completing the square, which dramatically simplifies the process of solving the quadratic equation.