Chapter 7.6, Problem 41WE

To determine

Re-write the equation $y=x^2+bx+c$ in the form $y-k=a{\left(x-h\right)}^2$ by completing square.

Answer to Problem 41WE

The equation is $y-\left(\frac{4c-b^2}{4}\right)={\left(x-\left(-\frac{b}{2}\right)\right)}^2$ .

Explanation of Solution

Given information: The given equation is,

  $y=x^2+bx+c\mathrm{....}\left(1\right)$

Calculation: The given equation $y=x^2+bx+c$ in $y-k=a{\left(x-h\right)}^2$ form is written as,

  $\begin{array}{c}y=x^2+bx+c\mathrm{....}\left(\text{From }\left(1\right)\right)\\ \text{or, }y-c=x^2+bx\\ \text{or, }y-c+{\left(\frac{b}{2}\right)}^2={\left(x\right)}^2+2\left(x\right)\left(\frac{b}{2}\right)+{\left(\frac{b}{2}\right)}^2\mathrm{....}\left(\text{Complete square}\right)\\ y-\left(\frac{4c-b^2}{4}\right)={\left(x-\left(-\frac{b}{2}\right)\right)}^2\end{array}$

Hence, the required equation is $y-\left(\frac{4c-b^2}{4}\right)={\left(x-\left(-\frac{b}{2}\right)\right)}^2$ .