Question

How do you solve $F(x) = 3{x^2} - 30x + 82$ into vertex form?

Answer 1

Hint : To solve this question, we have to recall the vertex form of an equation. Then, we use the given terms and manipulate them to apply the identity ${(a - b)^2} = {a^2} - 2ab + {b^2}$ in the equation, such that the remaining terms can be simplified to obtain the vertex form of the equation.

Complete step by step solution:
The vertex form of an equation is $a{(x - h)^2} + k$ .
We have the given expression as $F(x) = 3{x^2} - 30x + 82$ .
On analysing the expression, we notice that the identity ${(a - b)^2} = {a^2} - 2ab + {b^2}$ can be applied here, after adjusting the expression. To achieve this form,
We need to first take out a common factor for all of the term. We can do that as follows:
$F(x) = 3\left( {{x^2} - 10x + \dfrac{{82}}{3}} \right)$
Now, for identity, we need to split $\dfrac{{82}}{3}$ in two parts such that the value of one part is $25$ .
We can do that in this way, and simplify the expression further to obtain the vertex form of the equation.

$$F(x) = 3\left( {{x^2} - 10x + \dfrac{{75 + 7}}{3}} \right) \\\ \Rightarrow F(x) = 3\left( {{x^2} - 10x + 25 + \dfrac{7}{3}} \right) \\\ \Rightarrow F(x) = 3\left( {{x^2} - 10x + 25} \right) + 7 \\\ \Rightarrow F(x) = 3{\left( {x - 5} \right)^2} + 7 \;$$

Hence, the vertex form of $F(x) = 3{x^2} - 30x + 82$ is $F(x) = 3{\left( {x - 5} \right)^2} + 7$ .
So, the correct answer is “ $F(x) = 3{\left( {x - 5} \right)^2} + 7$ ”.

Note : The Vertex form is simply another form of a quadratic equation. The standard form of a quadratic equation is $a{x^2} + bx + c$ , while the vertex form of a quadratic equation is $a{(x - h)^2} + k$ . Here, $a$ is a constant that tells us whether the parabola opens upwards or downwards, and $(h,k)$ is the location of the vertex of the parabola. We cannot immediately read this from the standard form of a quadratic equation. Thus, vertex form is useful for solving quadratic equations, graphing quadratic functions, and more.