Factored Form Practice Problems - Quadratic Functions

We will begin by using the distributive property in order to expand the following expression.

(a+1)⋆(b+2) = ab+2a+b+2

We will then proceed to insert the known values into the equation and solve as follows:

(x-2)(x+5) =

x²-2x+5x+-2*5=

x²+3x-10

And that's the solution!

To find the standard representation of the quadratic function $f(x) = (x - 6)(x - 2)$, follow these steps:

• Step 1: Apply the FOIL method to expand the expression.
  - First: Multiply the first terms: $x \times x = x^2$.
  - Outside: Multiply the outer terms: $x \times (-2) = -2x$.
  - Inside: Multiply the inner terms: $(-6) \times x = -6x$.
  - Last: Multiply the last terms: $(-6) \times (-2) = 12$.
• Step 2: Combine the results from the FOIL method.
  - Combine all the expanded terms: $x^2 - 2x - 6x + 12$.
• Step 3: Simplify by combining like terms.
  - Combine the $x$-terms: $-2x - 6x = -8x$.
  - The expanded and simplified form is: $f(x) = x^2 - 8x + 12$.

By expanding and simplifying the given product, we have converted it to its standard form. Therefore, the standard representation of the function is $f(x) = x^2 - 8x + 12$.

The correct choice from the provided options is choice 2: $f(x) = x^2 - 8x + 12$.

To find the standard representation of the quadratic function given by $f(x) = (x+2)(x-4)$, we will expand the expression using the distributive property, commonly known as the FOIL method (First, Outer, Inner, Last):

• First: Multiply the first terms in each binomial: $x \cdot x = x^2$.
• Outer: Multiply the outer terms in the binomials: $x \cdot (-4) = -4x$.
• Inner: Multiply the inner terms: $2 \cdot x = 2x$.
• Last: Multiply the last terms in each binomial: $2 \cdot (-4) = -8$.

Now, let's combine these results:

The expression becomes $x^2 - 4x + 2x - 8$.

Next, we combine like terms:

The terms involving $x$ are $-4x + 2x$, which simplifies to $-2x$.

Thus, the expression simplifies to: $f(x) = x^2 - 2x - 8$

Upon comparing this result to the provided choices, we find that it matches choice 3.

Therefore, the standard representation of the function is $f(x) = x^2 - 2x - 8$.

To find the standard representation of the quadratic function $f(x) = 3x(x + 4)$, follow these steps:

• Step 1: Apply the distributive property to expand the expression $x(x + 4)$.
  Using this property, we have:
  $x(x + 4) = x \cdot x + x \cdot 4 = x^2 + 4x$.
• Step 2: Multiply each term by the coefficient outside the parenthesis, which is 3.
  This gives us:
  $3(x^2 + 4x) = 3 \cdot x^2 + 3 \cdot 4x$.
• Step 3: Simplify by performing the multiplication.
  $3x^2 + 12x$.

Therefore, the standard representation of the function is $f(x) = 3x^2 + 12x$. This matches choice 3 in the provided answers.

To solve this problem, we'll convert the given function from its factored form to the standard form using the distributive property. The given function is $f(x) = -x(x - 8)$.

Let's go through the necessary steps:

• Step 1: Apply the distributive property to expand the expression.
  $f(x) = -x(x - 8) = -x \cdot x + (-x) \cdot (-8)$
• Step 2: Simplify each term.
  $-x \cdot x = -x^2$ and $(-x) \cdot (-8) = +8x$.
• Step 3: Combine the terms to express $f(x)$ in standard form:
  $f(x) = -x^2 + 8x$.

Therefore, the standard representation of the function is $f(x) = -x^2 + 8x$.

Comparing this result to the multiple-choice options, we can see that the correct choice is option 3: $f(x)=-x^2+8x$.