Problem 69 - Algebra 1: Concepts and Skills

Question

Write the equation in standard form. $$ 6 x^{2}=5 x-7 $$

Step-by-Step Solution

Answer

The equation in standard form is \(6x^2 - 5x + 7 = 0\)

Step 1: Set the equation equal to zero

Subtract both sides of the equation with \(-5x\) and \(7\). This sets the equation to zero on right side. The equation becomes \(6x^2 - 5x + 7 = 0\)

Step 2: Check the standard form

Now, confirm that the equation is in standard form (\(Ax^2+Bx+C=0\)). The equation \(6x^2 - 5x + 7 = 0\) confirms to this standard form.

Key Concepts

Standard Form

The standard form of a quadratic equation is a way of writing it that is universally recognized. It is expressed as \( Ax^2 + Bx + C = 0 \). Here, \( A \), \( B \), and \( C \) are constants, where \( A \) is not equal to zero (since then it would not be a quadratic equation). This form shows all terms on one side and zero on the other, which makes it clear and simple to work with.

This uniform format is helpful in many scenarios, such as performing operations like factoring or using the quadratic formula to find solutions. When an equation is not in standard form, it is challenging to apply these techniques neatly. The coefficients \( A \), \( B \), and \( C \) must be real numbers to maintain the concept of a quadratic equation in standard form, ensuring a real solution if possible.

Setting the Equation to Zero

Setting an equation to zero is a fundamental step in solving quadratic equations. It involves rearranging all terms so that the equation equals zero, creating a balance on both sides. In our exercise, we start with the equation \( 6x^2 = 5x - 7 \).

• Firstly, move all the terms on one side of the equation—usually the left side.
• This means subtracting \( 5x \) and adding \( 7 \) to both sides of the equation.

As a result, the equation becomes \( 6x^2 - 5x + 7 = 0 \).

Reaching this "zero" state is critical because it allows us to solve the equation using methods such as factoring, completing the square, or applying the quadratic formula. It sets the groundwork for finding the roots or solutions of the quadratic equation.

Polynomial Equations

Polynomial equations involve expressions that include variables raised to whole number powers, with each term being a product of a constant and a single variable. Quadratic equations are a specific type of polynomial equation where the highest exponent of the variable is 2, hence the term "quadratic."

A quadratic equation typically has the form \( ax^2 + bx + c = 0 \), showcasing its polynomial nature. Each term in a polynomial equation consists of coefficients and variables. For example, in the equation \( 6x^2 - 5x + 7 = 0 \), the terms are ordered by decreasing power of \( x \):

• \( 6x^2 \) (quadratic term)
• \( -5x \) (linear term)
• \( + 7 \) (constant term)

Understanding the structure of polynomial equations is essential for identifying the type of equation and choosing appropriate solution strategies. Polynomial equations can represent various real-world phenomena, making them fundamental to mathematics and its applications.