Question

\(y=4(x-7)^2\)

Short answer

The expanded and simplified form of the given equation is \(y=4x^2 - 56x + 196\).

Step-by-step solution

Expand the equation

To begin, distribute the squared part of the equation. It will result in \(y=4(x^2 - 14x + 49)\)

Distribute the constant

Next, distribute the constant 4 to all terms within the parentheses: \(y=4x^2 - 56x + 196\)

Final form

The equation in its simplest form is then represented as \(y=4x^2 - 56x + 196\).

Key concepts

Expanding Expressions

Expanding expressions is a key step in simplifying algebraic equations, particularly when working with quadratic equations like the one given in the original exercise. Here, the expression was \(y=4(x-7)^2\), and our task was to expand it.

To start, remember that expanding involves reformulating an expression to get rid of parentheses. For a binomial squared, such as \( (x - 7)^2\), you apply the square on the binomial like so:

• First, write \( (x - 7)^2 \) as \( (x - 7)(x - 7) \).
• Next, use the distributive property to multiply \(x - 7 \) by itself.
• Multiply each term by each other: \( x \cdot x = x^2\), \( x \cdot (-7) = -7x\), and \((-7) \cdot (-7) = 49\).

After this expansion, we obtain the expression \(x^2 - 14x + 49\). This step sets the stage for applying further algebraic rules, such as the distributive property, to simplify the expanded form further.

Distributive Property

The distributive property is a foundational algebraic concept used to expand expressions and solve equations. In the context of this exercise, it was crucial for expanding and thereafter simplifying the quadratic expression.

The original equation \(y=4(x^2 - 14x + 49)\) requires distributing the constant multiplier — in this case, 4 — across each term of the expanded expression inside the parentheses. Here's how it works:

• Multiply \4 \ by \(x^2 \), resulting in \(4x^2\).
• Multiply \4 \ by \(-14x \), resulting in \(-56x\).
• Multiply \4 \ by \(49 \), resulting in \(196\).

This systematic application of the distributive property transforms the complete expression into \(y = 4x^2 - 56x + 196\). Each term is recalculated with the outside scalar, rendering the expression fully expanded and ready for interpretation as a quadratic equation in standard form.

Standard Form of a Quadratic Equation

Understanding the structure of a quadratic equation in its standard form helps in analyzing and solving quadratic problems. The standard form is written as \(ax^2 + bx + c = 0\), where:

• \(a\) represents the coefficient of the quadratic term \(x^2\),
• \(b\) is the coefficient of the linear term \(x\),
• \(c\) is the constant term.

In the original exercise, we've expanded and simplified the equation to \(y = 4x^2 - 56x + 196\). This expression is now in line with the standard form of a quadratic equation, although it defines a function \(y = f(x)\) rather than setting it to zero.

Identifying the standard form allows you to easily identify the values of \(a\), \(b\), and \(c\), which are essential parameters for further analysis such as finding the roots of the equation or defining its graph. Here, we have \(a = 4\), \(b = -56\), and \(c = 196\). Knowing the standard form is thus invaluable in mathematics for diagnosing and solving quadratic equations systematically.