Chapter 5.5, Problem 6CYU

To determine

To find: the value of c that makes trinomial a perfect square and write the trinomial as perfect square.

Answer to Problem 6CYU

The value of $\text{c=25}$ and perfect square is ${\left(\text{x-5}\right)}^2$ .

Explanation of Solution

Given:

  ${\text{x}}^{\text{2}}\text{-10x+c}$ .

Concept used:

An expression obtained from the square of binomial equation is a perfect square trinomial is in the form of ${\text{ax}}^{\text{2}}\text{+bx+c}$ is said to be a perfect square, if only if it satisfies the condition ${\text{b}}^{\text{2}}\text{=4ac}$ .

The perfect squares trinomial formula is given as:

  ${\left(\text{ax}\right)}^{\text{2}}{\text{+2abx+b}}^{\text{2}}\text{=}{\left(\text{ax+b}\right)}^{\text{2}}$ .

  ${\left(\text{ax}\right)}^{\text{2}}{\text{-2abx+b}}^{\text{2}}\text{=}{\left(\text{ax-b}\right)}^{\text{2}}$ .

Calculation:

  ${\text{x}}^{\text{2}}\text{-10x+c}$ .

To make the trinomial a perfect square, find the value of $\text{c}$ .

First Find the half of the coefficient of second term:

  $\frac{\text{-10}}{\text{2}}\text{=-5}$ .

Square the result of the half of the coefficient of the second term will be the $\text{c}$ :

  $\text{c=25}$ .

The trinomial ${\text{x}}^{\text{2}}\text{-10x+25}$ can be written as:

  $\Rightarrow {\text{x}}^{\text{2}}\text{-2}\left(\text{x}\right)\left(5\right)\text{+}{\left(5\right)}^2\text{-}{\left(5\right)}^2+25$ .

  $\Rightarrow {\left(\text{x-5}\right)}^2$ .

Hence the value of $\text{c=25}$ and perfect square is ${\left(\text{x-5}\right)}^2$ .

  ${\left(\text{x-3}\right)}^2+9=0$ .

  ${\left(\text{x-3}\right)}^2=-9$ .

  $\left(\text{x-3}\right)=\pm \sqrt{-9}$ .

  $\left(\text{x-3}\right)\text{=±3i}$

  $\text{x=3i+3}$ and $\text{x=-3i+3}$ .