Question

Write the equation $y=3x^2-6x+5$ in the form$y=a{\left(x-h\right)}^2+ky=a{\left(x-h\right)}^2+k$.

Short answer

The required form of the equation in$y=a{\left(x-h\right)}^2+k$is $y=3{\left(x-1\right)}^2+2$ where $a=3,h=1\text{and}k=2$.

Step-by-step solution

Step 1. Write down the given information.

The given equation is $y=3x^2-6x+5$.

Step 2. Calculation.

The given equation$y=3x^2-6x+5$is converted to $y=a{\left(x-h\right)}^2+k$form as:

$\begin{array}{c}y=3x^2-6x+5\\ =3\left(x^2-2x+\frac{5}{3}\right)\\ =3\left(x^2-2x+\frac{5}{3}+{\left(-1\right)}^2-{\left(-1\right)}^2\right)\mathrm{....}\left(\text{Addandsubtract}{\left(-1\right)}^2\right)\\ =3\left(\left(x^2-2x+{\left(-1\right)}^2\right)+\left(\frac{5}{3}-1\right)\right)\\ =3\left({\left(x-1\right)}^2+\frac{2}{3}\right)\\ =3{\left(x-1\right)}^2+2\end{array}$

Here $a=3,h=1\text{and}k=2$.

Step 3. Conclusion.

The required form of the given equation is$y=3{\left(x-1\right)}^2+2$where $a=3,h=1\text{and}k=2$.