Q59. - Algebra 2

Question

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

Step-by-Step Solution

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 $a=-3, h=-3 \text{and }k=17$.

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:

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

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

Step 3. Conclusion.

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