Question

Write the equation$y=-3x^2-18x-10$in the form $y=a{\left(x-h\right)}^2+k$.

Short answer

The required form of the equation in$y=a{\left(x-h\right)}^2+k$is $y=-3{\left(x+3\right)}^2+17$where localid="1648620480283" $a=-3,h=-3\text{and}k=17$.

Step-by-step solution

Step 1. Write down the given information.

The given equation is$y=-3x^2-18x-10$.

Step 2. Calculation.

The given equation $y=-3x^2-18x-10$is converted to $y=a{\left(x-h\right)}^2+k$form as:

$\begin{array}{c}y=-3x^2-18x-10\\ =-3\left(x^2+6x+\frac{10}{3}\right)\\ =-3\left(x^2+6x+\frac{10}{3}+{\left(3\right)}^2-{\left(3\right)}^2\right)\mathrm{....}\left(\text{Addandsubtract}{\left(3\right)}^2\right)\\ =-3\left(x^2+6x+{\left(3\right)}^2+\left(\frac{10}{3}-9\right)\right)\\ =-3\left({\left(x+3\right)}^2-\frac{17}{3}\right)\\ =-3{\left(x+3\right)}^2+17\end{array}$

Here localid="1648620544915" $a=-3,h=-3\text{and}k=17$.

Step 3. Conclusion.

The required form of the given equation is $y=-3{\left(x+3\right)}^2+17$where localid="1648620558780" $a=-3,h=-3\text{and}k=17$.