Question

Complete the square for each quadratic function. $$ f(x)=x^{2}-10 x+7 $$

Short answer

(x - 5)^2 - 18

Step-by-step solution

Identify coefficients

In the quadratic function, identify the coefficients of the terms. Here, the quadratic function is given as \( f(x) = x^2 - 10x + 7 \) . The coefficient of the linear term \( x \) is \( -10 \) and the constant term is \( 7 \).

Formulate the complete square expression

To complete the square, start by focusing on the quadratic and linear terms. Write down the expression: \( x^2 - 10x\) . Next, we can add and subtract the square of half the coefficient of \( x \) . For this expression, it gives \( \left( - \frac{10}{2} \right)^2 = (-5)^2 = 25 \) .

Add and subtract the square inside the function

Insert the square value identified in the previous step inside the function, then subtract it outside to keep the equation balanced: \( f(x) = x^2 - 10x + 25 - 25 + 7 \).

Simplify by factoring

Group the perfect square trinomial and simplify the equation. Now, we can rewrite it as follows: \( f(x) = (x - 5)^2 - 18 \) . This is the completed square form.

Key concepts

Quadratic function

A quadratic function is a type of polynomial that has a degree of 2, meaning the highest exponent of the variable (usually x) is 2. This type of function generally looks like:

$$ f(x) = ax^2 + bx + c $$

Here, the terms $$ ax^2 $$ for the quadratic term, $$ bx $$ for the linear term, and the constant term $$ c $$ describe its structure. Quadratic functions graph as parabolas, which can open upward or downward based on the sign of the leading coefficient $$ a $$.

Let's dive into the specifics of the equation given in the original problem: $$ f(x) = x^2 - 10x + 7 $$.

The task is to complete the square for this quadratic function to rewrite it into a form that reveals the vertex of the parabola more clearly.

Coefficients

The coefficients in a quadratic function are crucial as they influence the shape and position of the parabola on a graph. In the expression $$ f(x) = x^2 - 10x + 7 $$, we have the following coefficients:

• The coefficient of $$ x^2 $$ is 1 (though it's not explicitly written).
• The coefficient of $$ x $$ is -10.
• The constant term (the number on its own) is 7.

Identifying these coefficients correctly is the first step in methods like completing the square or factoring the quadratic expression. Notice that in the solution, we focus on the linear and quadratic terms first to simplify and reframe the equation.

Perfect square trinomial

A perfect square trinomial is a special kind of quadratic expression that can be factored into a binomial square. For example: $$ (x - 5)^2 $$ expands back to $$ x^2 - 10x + 25 $$.

In solving our equation, we create the perfect square trinomial by adding and subtracting the same number. Here's how:

First, consider only $$ x^2 - 10x $$, then add and subtract the square of half the linear coefficient: $$ (- \frac{10}{2})^2 = (-5)^2 = 25 $$.

This gives us: $$ x^2 - 10x + 25 - 25 + 7 $$, which can be regrouped as: $$ (x - 5)^2 - 18 $$.

By completing the square, we convert the quadratic expression into something that visually represents a shifted and vertically translated parabola.

Factoring

Factoring involves breaking down a complex expression into simpler terms (or factors) that, when multiplied together, yield the original expression. Completing the square is closely related to factoring. In the context of our quadratic function: $$ f(x) = (x - 5)^2 - 18 $$,
we observe that $$ (x - 5)^2 $$ is a factored form of our perfect square trinomial.

This step is critical because it allows us to express the quadratic function in a form that's easy to analyze and graph. Factoring simplifies solving equations and finding the function's roots. In our case, we achieve this by converting the original terms into $$ (x - 5)^2 $$ - a clear, factored, and completed square form.