Problem 21 - Elementary Algebra

Question

For the following problems, write the values of \(a, b,\) and \(c\) in quadratic equations. $$ 5 x^{2}-3 x+9=4 x^{2} $$

Step-by-Step Solution

Answer

Answer: In the given quadratic equation, the values are \(a=1\), \(b=-3\), and \(c=9\).

Step 1: Rewrite the given equation in the general form of a quadratic equation

First, we need to rewrite the given equation in the general form of a quadratic equation. We do this by moving all terms to one side of the equation. The given equation is: $$ 5x^2 - 3x + 9 = 4x^2 $$ Now, subtract \(4x^2\) from both sides to get: $$ 5x^2 - 4x^2 - 3x + 9 = 0 $$

Step 2: Simplify the equation

Now we will simplify the equation by combining like terms. In this case, we can combine the \(5x^2\) and \(-4x^2\) terms. So the simplified equation is: $$ (5x^2 - 4x^2) -3x + 9 = x^2 - 3x + 9 = 0 $$

Step 3: Identify the values of 'a', 'b', and 'c'

Now that the equation is in the general form, we can identify the coefficients 'a', 'b', and 'c' by comparing it to the general form \(ax^2 + bx + c = 0\). In our simplified equation, \(x^2 - 3x + 9 = 0\), we have: \(a=1\), since there's an implied \(1\) in front of the \(x^2\) term \(b=-3\), since this is the coefficient for the \(x\) term \(c=9\), since this is the constant term in the equation So the values of \(a\), \(b\), and \(c\) in the given quadratic equation are \(a=1\), \(b=-3\), and \(c=9\).

Key Concepts

Algebra

Algebra is a branch of mathematics that often represents numbers with symbols, allowing us to solve a variety of problems from the simple to the complex. This discipline introduces us to concepts such as variables, expressions, and equations that form the foundation for understanding more advanced mathematical concepts.

At the heart of algebra is the idea of using letters to represent numbers. For instance, in the quadratic equation \(ax^2 + bx + c = 0\), the variables \(a\), \(b\), and \(c\) represent coefficients, and \(x\) is the variable for which we're often trying to solve. One of the first skills algebra students master is the ability to manipulate these equations to isolate and solve for the unknown variable. This is done by applying various algebraic rules and operations such as addition, subtraction, multiplication, division, and factoring.

Solving Quadratics

Solving quadratic equations is a fundamental skill in algebra. A quadratic equation is a second-degree polynomial, typically in the form \(ax^2 + bx + c = 0\), and can curve into a parabola when graphed. The challenge with quadratics is finding the values of \(x\) that satisfy the equation, also known as the roots of the equation.

To solve quadratics, one can use various methods such as factoring, completing the square, the quadratic formula, or graphing. The choice of method often depends on the structure of the given quadratic. For example, factoring is often the quickest way to solve a quadratic, but it only works when the quadratic can be expressed as a product of two binomials. The quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), on the other hand, can solve any quadratic equation, provided the discriminant \(\sqrt{b^2 - 4ac}\) is not imaginary or involves complex numbers.

Coefficients

In the context of quadratic equations, the term 'coefficient' refers to the numerical factor of each term. The coefficients in the equation \(ax^2 + bx + c = 0\) determine the shape and position of the parabola when graphed. Here's a closer look at what each coefficient represents:

• \(\mathbf{a}\): The leading coefficient affects the parabola's width and direction. If \(a > 0\), the parabola opens upwards, and if \(a < 0\), it opens downwards. The absolute value of \(a\) indicates how 'wide' or 'narrow' the parabola will be—the larger the value, the narrower the parabola.
• \(\mathbf{b}\): This coefficient affects the parabola's lateral position on the graph. It is part of the term that determines the vertex, along with \(a\).
• \(\mathbf{c}\): Known as the constant term, it affects the vertical position of the parabola, particularly where it intersects the \(y\)-axis.

Identifying and understanding the role of these coefficients is crucial as they provide insights into the graph's behavior without needing to plot it. Additionally, knowing the coefficients helps us to apply the quadratic formula and other solution methods effectively.