Question #8950d

Jun 23, 2017

$y = {\left(x - 2.5\right)}^{2} + 94.75$

Explanation:

$y = {x}^{2} - 5 x + 101$

To complete the square, your goal is to make a perfect trinomial. For example:

${x}^{2} + 2 x + 1 = {\left(x + 1\right)}^{2}$

To make things easier, let's add some parentheses to the equation.

$y = \left({x}^{2} - 5 x\right) + 101$

You need to add a constant into the parentheses to create a perfect trinomial. To find this constant, use this formula:

${\left(\frac{b}{2}\right)}^{2}$

The $b$ comes from the standard form of a quadratic equation $y = a {x}^{2} + b x + c$. In this case, that means your $b$ is $- 5$.

${\left(- \frac{5}{2}\right)}^{2}$

${\left(- 2.5\right)}^{2}$

$6.25$

This is where things can get a little confusing. Because you are adding a number on one side of the equation, and not the other side, you need to balance it out on the same side.

$y = \left({x}^{2} - 5 x + 6.25\right) + 101 - 6.25$

As you can see, you've created your perfect trinomial while also not unbalancing your equation. Now you can factor and simplify!

$y = {\left(x - 2.5\right)}^{2} + 101 - 6.25$

$y = {\left(x - 2.5\right)}^{2} + 94.75$

Jun 23, 2017

$y = {\left(x - \frac{5}{2}\right)}^{2} + \frac{379}{4}$

Explanation:

${x}^{2} - 5 x + 101$

To complete the square, we need to find a value that makes ${x}^{2} - 5 x$ a perfect square. To make it easier to visualize, I like to move the other component ($101$) to the other side of the equation.

$- 101 = {x}^{2} - 5 x$

To solve for the missing component, we need to follow these steps:

1) take the middle term, $- 5$ and divide by $2$

$\frac{- 5}{2} = - \frac{5}{2}$

2) square this solution

${\left(- \frac{5}{2}\right)}^{2} = \frac{25}{4}$

Now, let's add this to the equation. REMEMBER in an equation, we can add whatever we want, but we must also add it to the other side:

$- 101 + \frac{25}{4} = {x}^{2} - 5 x + \frac{25}{4}$

$- \frac{379}{4} = {\left(x - \frac{5}{2}\right)}^{2}$

${\left(x - \frac{5}{2}\right)}^{2} + \frac{379}{4}$

Now we have our solution!

$y = {\left(x - \frac{5}{2}\right)}^{2} + \frac{379}{4}$