Quadratic Equation
A quadratic equation is a type of algebraic equation that involves an unknown variable, typically represented as \(x\), raised to the second power, which means it has an exponent of 2. The standard structure of a quadratic equation is characterized by three terms: the quadratic term \(ax^2\), the linear term \(bx\), and the constant term \(c\). An essential characteristic of these equations is that they can have up to two real solutions, which are the values of \(x\) that make the equation true.
When we look at the exercise \(10x^2=90\), we identify it as a quadratic because of the squared term. The goal is to transform this equation into a recognizable format where we can apply various methods to find these solutions. Grasping the nature of quadratic equations is fundamental as they appear frequently in various disciplines including physics, engineering, finance, and beyond.
General Form of a Quadratic
The general form of a quadratic equation is \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are coefficients, and \(a\) is not equal to zero. If \(a\) were zero, the equation would no longer be quadratic but linear. This form is widely used because it sets the stage for applying several methods to solve for \(x\), such as factoring, completing the square, or using the quadratic formula.
In the given exercise, transforming \(10x^2=90\) into the general form involves normalizing the quadratic term so it would ideally have a coefficient of 1 and moving all terms to one side to set the equation to zero. The resulting general form of this specific quadratic equation is \(x^2 - 9 = 0\), which elegantly showcases the required structure of a quadratic equation in general form.
Algebraic Equations
Algebraic equations are mathematical statements that assert the equality of two algebraic expressions. These expressions can involve variables, coefficients, and constants, and can range from simple, linear equations to more complex polynomials. The act of solving an algebraic equation entails finding the value or values of the variable that make the equation true.
Solving algebraic equations is like unraveling a puzzle, requiring a sequential application of algebraic rules and operations such as addition, subtraction, multiplication, division, and factoring. Mastery of these equations is crucial because they serve as the foundation for more advanced topics in algebra and calculus.
Solving Quadratics
Solving quadratics is the process of finding the values of \(x\) that satisfy the quadratic equation \(ax^2 + bx + c = 0\). There are various methods to approach this, each appropriate for different forms of quadratic equations.
Factoring
Factoring is often the first method tried, especially if the quadratic easily breaks down into two binomials. However, not all quadratics are factorable using integers, which is where other methods come in.
Completing the Square
This method involves creating a perfect square trinomial from the quadratic expression, making it easier to solve for \(x\).
Quadratic Formula
For any quadratic equation, the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), will find the solutions, provided that \(a\) is not zero.
Graphical Methods
Visual learners might prefer these, as they involve drawing the quadratic as a parabola and identifying the points where it crosses the x-axis. Each method requires different steps and presents its own learning opportunities. To become proficient in solving quadratics, students should practice all these methods and understand when each is most effective.
