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Expanding expressions is a fundamental skill in algebra, especially important when working with quadratic equations. When you see an expression like

• \((x+4)^2\),

your goal is to remove the parentheses to find its equivalent expanded form. Here, expanding means simply the removal of parentheses to express each component separately using arithmetic procedures - particularly the distributive property.

To expand

• \((x+4)^2\),

you recognize it as

• \((x+4)(x+4)\).

Use the distributive property by multiplying each term in the first expression by each term in the second. This gives you two separate calculations:

• \(x(x+4)\) and
• \(4(x+4)\).

Apply the distributive property to get

• \(x^2 + 4x\) from the first calculation and
• \(4x + 16\) from the second.

Finally, combine these results to get

• \(x^2 + 8x + 16\).

Expanding expressions is straightforward once you consistently apply these steps.

Quadratic equations often appear in different forms, each offering unique insights into the function's properties. The vertex form is one such expression,

• \(y=a(x-h)^2+k\),

which emphasizes the vertex of the parabola.

In our example,

• \(y = (x+4)^2\),

the equation is already in vertex form. Here,

• h = -4
• and k = 0.

This means the vertex of the parabola is at

• \((-4,0)\).

Vertex form is particularly useful when you need to quickly identify the maximum or minimum point of a quadratic function, known as the vertex. This is because it also reveals whether the parabola opens upwards

• \(a>0\)

or downwards

• \(a<0\).

Understanding vertex form allows you to easily manipulate and graph quadratic functions, making it indispensable for solving problems involving parabolas.

The general form of a quadratic equation shows its most expanded polynomial state. It's written as

• \(y=ax^2+bx+c\).

This form is useful for solving quadratic equations using the quadratic formula or determining the roots.

To convert from vertex to general form, we expand

• \((x+4)^2\) as described earlier,

arriving at

• \(x^2+8x+16\).

This means in the expression,

• \(a=1\),
• \(b=8\),
• and \(c=16\).

The coefficients correspond respectively to each term of the quadratic, linear, and constant elements in the expression. General form makes it easy to apply further mathematical techniques, such as factoring or completing the square, and is essential for considering the full scope of operations you can perform on a quadratic equation.