Express x^{2}-5x+\dfrac {1}{4} in the form \left( x-a\right) ^{2}-b

Answer:

\left( x-a\right) ^{2}-b is equal to x^{2}-2ax+a^{2}-b

Expanding [x^{2}-5x)] we get [x^{2}-2 \times x \times \dfrac{5}{2}+ {(\dfrac{5}{2})}^2] - {(\dfrac{5}{2})}^2

\therefore x^{2}-5x+\dfrac {1}{4} can be written as

[x^{2}-2 \times x \times \dfrac{5}{2}+ {(\dfrac{5}{2})}^2] - {(\dfrac{5}{2})}^2+\dfrac {1}{4}

or ({x - \dfrac{5}{2})}^{2} - \dfrac{25}{4}+ \dfrac{1}{4}

or ({x - \dfrac{5}{2})}^{2} - \dfrac{25-1}{4}

or ({x - \dfrac{5}{2})}^{2} - \dfrac{24}{4}

or ({x - \dfrac{5}{2})}^{2} - 6

\therefore x^{2}-5x+\dfrac {1}{4} can be expressed the form \left( x-a\right) ^{2}-b as

\bold{({x - \dfrac{5}{2})}^{2} - 6} where a=\dfrac{5}{2} and b=6

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