Question 153·Hard·Equivalent Expressions

After you have the simplified quadratic, factor out the coefficient of $x^2$ from the $x^2$ and $x$ terms so that you have something like $k(x^2 + \text{(something)}x) + \text{constant}$.

Inside the parentheses, complete the square: remember that $(x - a)^2 = x^2 - 2ax + a^2$. Use this pattern to rewrite the quadratic inside the parentheses, and then adjust the constant term outside so the expression stays equivalent.

Once you have an expression of the form $k(x - m)^2 + n$, carefully identify $k$, $m$, and $n$ and then add them together.

In Desmos, type f(x) = 4x^2 - 40x + 101 + (x - 5)^2 to graph the original quadratic function.

On a new line, type g(x) = a(x - b)^2 + c. Desmos will create sliders for a, b, and c, which play the roles of $k$, $m$, and $n$.

Adjust the sliders for a, b, and c until the graph of g(x) lies exactly on top of the graph of f(x). The values of a, b, and c at that point correspond to $k$, $m$, and $n$.

Once the graphs match, read off the slider values for a, b, and c, then calculate $k + m + n$ from those numbers on your own (for example, using the Desmos calculator line if you wish).

Start by expanding $(x - 5)^2$ and then combining like terms.

• Expand:
  • $(x - 5)^2 = x^2 - 10x + 25$

Substitute this back into the expression:

Now combine like terms:

• $4x^2 + x^2 = 5x^2$
• $-40x - 10x = -50x$
• $101 + 25 = 126$

So the expression becomes:

To match the form $k(x - m)^2 + n$, first factor out the coefficient of $x^2$, which is 5, from the terms involving $x$:

Now you will complete the square inside the parentheses.

Focus on $x^2 - 10x$ inside the parentheses.

To complete the square:

• Take half of the coefficient of $x$: half of $-10$ is $-5$.
• Square it: $(-5)^2 = 25$.

So,

(Subtract 25 to keep the expression equivalent.)

Substitute this back into the factored form:

Now distribute the 5:

Combine the constants:

Now the expression is written as:

This matches $k(x - m)^2 + n$ with:

• $k = 5$
• $m = 5$
• $n = 1$

Now compute $k + m + n$:

So, the value of $k + m + n$ is $11$.