Step 1
Given;
[tex]x^2-5x+4[/tex]Required; To write it in the form below
[tex](x-a)^2+b[/tex]Step 2
Use the method of completing the square to rewrite the expression as required.
The steps to completing the square method include;
1) Divide all terms by the coefficient of x² especially if it is not 1
2) Separate the expression into 2 brackets. The constant term will be in 1 bracket and the other terms in another bracket.
3)Find half of the coefficient of x and add the square of half of the coefficient of x to the bracket containing the other terms but subtract it from the constant term.
4)Simplify the square of half of the coefficient of x on both sides
5) Simplify the constant term by adding up or subtracting as the case may be.
6) Factorize the other bracket containing the square of half of the coefficient of x simplified, x², and x to get the two factors that must be identical binomials as shown below
7) Write the factorized perfect square or identical binomials and the simplified constant sides together to get the answer in the required form.
[tex]\begin{gathered} ((x^2-5x)+6.25)+(4-6.25) \\ ((x^2-5x)+6.25)+(-2.25) \\ (x^2-5x+6.25)+(-2.25) \\ (x^2-5x+6.25)\text{ is a perfect square} \\ A\text{ p}\operatorname{erf}ect\text{ }square\text{ is a }quadratic\text{ expression that factors out to identical binomials} \\ \text{Therefore,} \\ We\text{ will factorize it searching for 2 factors such;} \\ \text{When we add them we get -5x.} \\ ^{}\text{When we multiply them we get 6.25} \end{gathered}[/tex]These factors are -2.5x and -2.5x
We will now replace -5x with the 2 factors (-2.5x - 2.5x)
Note; These two factors still sum up to -5x and when multiplied give 6.25
[tex]\begin{gathered} (x^2-5x+6.25) \\ (x^2-2.5x-2.5x+6.25) \\ (x^2-2.5x)(-2.5x+6.25) \\ \text{Get the GCF in both brackets} \\ \text{GCF}=x\text{ in the first bracket and -2.5 in the second bracket} \\ x(\frac{x^2}{x}-\frac{2.5x}{x})-2.5(-\frac{2.5x}{-2.5}+\frac{6.25}{-2.5}) \\ x(x-2.5)-2.5(x-2.5) \\ \text{Hence,} \\ we\text{ have; (x-2.5)(x-2.5)} \\ \text{(x-2.5)(x-2.5)}=(x-2.5)^2 \end{gathered}[/tex]Hence, we will have;
[tex](x-2.5)^2_{}+(-2.25)[/tex]Hence the given equation, written in the form of (x-a)²+b will be;
[tex](x-2.5)^2_{}+(-2.25)[/tex]