Problem 20 - Algebra 1: Concepts and Skills

Question

Write the equation in standard form. Identify the values of a, b, and c. $$-2 t^{2}=-8$$

Step-by-Step Solution

Answer

The equation in standard form is \(-2t^2 + 8 = 0\). The values of a, b, and c are -2, 0, and 8 respectively.

Step 1: Rewrite Equation in Standard Form

To write the equation in standard form, it needs to be structured as \(ax^2 + bx + c = 0\). Currently, the given equation is \(-2t^2 = -8\). By moving the -8 to the left side of the equation, the standard form would be \(-2t^2 + 8 = 0\).

Step 2: Identify the Coefficients

Now that the equation is noted as \(-2t^2 + 8 = 0\), the coefficients (a, b, and c) can be identified. Here 'a' is the coefficient of \(t^2\), 'b' is the coefficient of 't', and 'c' is the constant term. From the equation you can observe that the coefficient 'a' is -2, there is no 't' term so 'b' is 0, and 'c' is 8.

Key Concepts

Quadratic Equations

Quadratic equations are fundamental in algebra and appear in the form of a polynomial equation of the second degree. This means they have the highest exponent as 2. The general structure can be expressed as \(ax^2 + bx + c = 0\), where:

• \(a\), \(b\), and \(c\) are coefficients
• \(x\) is the variable
• \(a eq 0\) because if \(a\) were zero, the equation would not be quadratic but linear.

Quadratic equations are often used to model various real-world phenomena such as projectile paths and areas. They are solved using different methods such as factoring, completing the square, or using the quadratic formula. The term \(ax^2\) is known as the quadratic term, \(bx\) as the linear term, and the constant \(c\) does not contain the variable and is simply a number. Recognizing these parts is essential when dealing with these types of equations.

Coefficients Identification

Identifying coefficients in a quadratic equation is a straightforward process but essential for solving the equation. Coefficients are the numerical factors associated with the variables in polynomial expressions.
In the standard quadratic form, \(ax^2 + bx + c = 0\):

• The coefficient \(a\) is attached to the \(x^2\) term. It determines the parabola's direction and width.
• The coefficient \(b\) relates to the \(x\) term. If there is no \(x\) term, as in some equations, \(b\) is considered to be zero.
• The constant \(c\) is the free term that doesn't involve \(x\). This number affects the position of the graph on the y-axis.

In the provided example, the equation \(-2t^2 + 8 = 0\) shows:

• \(a = -2\), indicating the parabola opens downward as it is negative.
• \(b = 0\), since there is no \(t\) term.
• \(c = 8\), the constant term.

Clear identification of these coefficients is crucial when applying any solving method or graph interpretation.

Polynomial Expressions

Polynomial expressions are combinations of variables, coefficients, and constants. They can have one or multiple terms.
The terms in a polynomial expression are separated by addition or subtraction. Each term consists of a constant multiplied by a variable raised to a power. The general form of a polynomial includes terms like \(an\cdot x^n\), where:

• \(an\) is the coefficient
• \(x\) is the variable
• \(n\) is the exponent

A quadratic expression is specifically a type of polynomial with the highest power of 2.
In the expression \(-2t^2 + 8\), which was obtained by arranging the original equation into standard form, the polynomial part \(-2t^2 + 0t + 8\) involves:

• The quadratic term \(-2t^2\).
• The linear term \(0t\), which is unseen in the original form due to its coefficient being zero.
• The constant term \(+8\).

Understanding polynomial expressions helps simplify and solve Algebra problems effectively. This knowledge is often built upon to tackle more complex expressions or equations.