Problem 74 - Algebra 1: Concepts and Skills

Question

Write the equation in standard form. $$ 8=5 x^{2}-4 x $$

Step-by-Step Solution

Answer

The equation in standard form is \(x^{2} - \frac{4}{5}x - \frac{8}{5} = 0\)

Step 1: Write the Equation in general form

Rewrite the given equation with the \(x\) terms on one side and the constant on the other: \(5x^{2}-4x-8 = 0\)

Step 2: Identify Coefficients

From the equation \(5x^{2}-4x-8 = 0\), identify \(a = 5\), \(b = -4\), and \(c = -8\)

Step 3: Make the Coefficient of the Quadratic Term 1

Transform the equation such that the coefficient of \(x^2\) becomes 1. To achieve that, divide the whole equation by \(a = 5\), resulting in \(x^{2} - \frac{4}{5}x - \frac{8}{5} = 0\)

Key Concepts

Convert Quadratic Equation

Converting a quadratic equation involves changing its format without altering its solutions. Essentially, you are reorganizing the components of the equation to achieve a more standard form. Begin by examining the original equation. In this case:

• Start with the equation: \[ 8 = 5x^2 - 4x \]

To convert it into a more recognizable quadratic format, bring all the terms to one side, generally placing the variable terms on one side and setting the equation to zero:

• First subtract 8 from both sides: \[ 5x^2 - 4x - 8 = 0 \]

This step aligns the equation with the general format of quadratic equations.

Coefficients in Quadratic Equation

Understanding the coefficients in a quadratic equation is crucial, as they determine the shape and position of the parabola represented by the equation. In the standard form of a quadratic equation, which we write as \( ax^2 + bx + c = 0 \), the coefficients are:

• a: The coefficient of \( x^2 \), determining the direction and width of the parabola.
• b: The coefficient of \( x \), influencing the parabola's position horizontally.
• c: The constant term, which shifts the parabola up or down the y-axis.

Referring back to the standard form we derived: \[ 5x^2 - 4x - 8 = 0 \], you identify:

\( a = 5 \)

\( b = -4 \)

\( c = -8 \)

These numbers influence the curve and intersections of the graph, playing a vital role in quadratic calculations.

Quadratic Equation Standard Form

The standard form of a quadratic equation is a tidy way to express quadratic expressions, crucial for graphing and solving. Typically, we rearrange quadratic equations to appear as:

• \[ ax^2 + bx + c = 0 \]

This format is especially useful because it allows us to utilize several mathematical tools, like the quadratic formula, to find solutions. By transforming our given equation \[ 5x^2 - 4x - 8 = 0 \], we notice that it already mirrors the standard form. However, if needed, you can simplify the equation by making the coefficient of \( x^2 \) equal to one, by dividing all terms by the leading coefficient \( a = 5 \):

• \[ x^2 - \frac{4}{5}x - \frac{8}{5} = 0 \]

This is often done to simplify further calculations or graphing. Being familiar with these transformations helps efficiently handle quadratic equations.