Question

Write the quadratic equation in general form. $$ x(x+2)=3 x^{2}+1 $$

Short answer

The quadratic equation in general form is \(0 = x^2 - x + 0.5\).

Step-by-step solution

Expand the left side of the equation

The first step is to expand \(x(x+2)\). It is done by multiplying \(x\) by each term in the bracket, yielding \(x^2 + 2x\). So our equation now becomes: \(x^2 + 2x = 3x^2 + 1\). This step involves the distributive property of multiplication over addition.

Rearrange to the general form of a quadratic equation

The next step is to rearrange the equation such that all terms are on one side, resulting in a quadratic in general form. We can get there by subtracting \(x^2\) and \(2x\) from both sides of the equation. Our equation then becomes: \(0 = 3x^2 - x^2 + 1 - 2x\). This simplifies to: \(0 = 2x^2 -2x +1\).

Final simplification

Finally, for the sake of simplicity, we can divide the entire equation through by the coefficient of \(x^2\), which is 2, to get: \(0 = x^2 - x + 0.5\). This is our quadratic equation in general form, with \(a=1\), \(b=-1\), and \(c=0.5\). The quadratic is therefore now completely simplified and in the correct form.

Key concepts

General Form of a Quadratic Equation

The general form of a quadratic equation is an algebraic expression that is structured as \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants with \(a \eq 0\). This is because if \(a = 0\), the equation would no longer be quadratic but linear. In the context of the exercise given, \(x(x+2)=3x^{2}+1\), the goal is to manipulate the equation to find its general form, a process which involves expanding, rearranging, and simplifying terms.

The significance of a quadratic equation being in this form lies in the fact that it is ready for various methods of solutions, such as factoring, completing the square, or using the quadratic formula. It also allows for easy identification of the parabola's direction of opening when graphed and can be analyzed to find its vertex, axis of symmetry, and intercepts.

Distributive Property of Multiplication

The distributive property of multiplication over addition is a fundamental concept that allows for the simplification of equations. This property states that for any three numbers, say \(a\), \(b\), and \(c\), the equation \(a(b + c) = ab + ac\) holds true. In the exercise \(x(x + 2) = 3x^2 + 1\), this property is applied to expand the left side of the equation.

By distributing \(x\) across the terms inside the parentheses, we obtain \(x^2 + 2x\). This step is essential to transforming the equation into a recognizable quadratic form, which then can be manipulated further. It's crucial for students to understand that the distributive property is what allows us to change the appearance of an equation without changing its meaning, paving the way to achieve the desired form for further analysis.

Simplifying Quadratic Equations

Simplifying a quadratic equation involves reducing it to its most basic form without changing its solutions. This process may include combining like terms, which are terms that have the same variables raised to the same power, and simplifying constants.

In the exercise, once we reach the equation \(0 = 3x^2 - x^2 + 1 - 2x\), simplifying is done by combining the \(x^2\) terms and the \(x\) terms to yield \(0 = 2x^2 - 2x + 1\). This version is more streamlined but still maintains the integrity of the original equation. Simplifying allows for a clearer view of the equation’s components, aiding in problem-solving strategies such as factoring or application of the quadratic formula.

Rearranging Quadratic Equations

Rearranging a quadratic equation refers to the process of moving all of the terms to one side of the equal sign, typically resulting in a zero on the other side. This step is pivotal in obtaining the general form \(ax^2 + bx + c = 0\). After expanding and simplifying the given exercise equation, we subtract terms from both sides to collect all variable terms to one side, resulting in \(0 = 2x^2 - 2x + 1\).

Rearranging makes the structure of the equation suitable for further analysis and application of solution methods. It's a preparatory step, much like tidying up the workspace before beginning a complex task, and its necessity cannot be overstated for students grappling with solving quadratic equations.