Question

How do you write $ f(x) = 3{x^2} - 6x + 9 $ into vertex form.

Answer 1

Hint: As we know that the above equation is a quadratic equation. The standard form of the quadratic form is $ a{x^2} + bx + c $ . We know that the vertex form of the quadratic equation is $ y = a{(x - h)^2} + k $ , where $ (h,k) $ is the vertex of the equation. We will first find the coordinates of the vertex and then we put them in their formula.

Complete step by step solution:
Here we have an equation: $ 3{x^2} - 6x + 9 $ . Now we know that the $ x $ - coordinate of the vertex is $ x = \dfrac{{ - b}}{{2a}} $ , where the value of $ b = - 6 $ and $ a = 3 $ .
By putting the values in the formula we have $ \dfrac{{ - b}}{{2a}} = \dfrac{{ - ( - 6)}}{{2 \times 3}} $ . It gives the value of x-coordinate i.e. $ \dfrac{6}{6} = 1 $ .
Now by putting the value of $ x $ in the quadratic equation we have $ 3(1) - 6(1) + 9 $ .
It gives the value of y-coordinate i.e. $ 6 $ . Since the vertex form of the quadratic equation is $ y = a{(x - h)^2} + k $ , therefore $ y = 3{(x - 1)^2} + 6 $ .
Hence the vertex form of $ 3{x^2} - 6x + 9 $ is $ 3{(x - 1)^2} + 6 $ .
So, the correct answer is “$ 3{(x - 1)^2} + 6 $”.

Note: We should keep in mind while solving this kind of quadratic equation that we use correct formulas or values to find the vertex form and keep checking the negative and positive sign otherwise it will give the wrong answer. Also we should always check for the sum and product and also verify the factors by multiplying that as it will provide the same above quadratic equation or not. These are some of the standard algebraic identities.