Question

How do you solve `y = x^2 − 14x + 24` graphically and algebraically?

Answer 1

There are 3 ways to solve this quadratic algebraically. The most simple way to solve your problem would be to use the product and sum method.

Explanation:

Before we begin, another way to write `y=x^2 - 14x^2 + 24` is
`x^2 - 14x^2 + 24=0`. This makes it easier as it is now a written as a simple quadratic, ready to solve.

So, firstly, the product and sum of `y=x^2 - 14x^2 + 24` is:
Product =24
Sum = -14

Now the next step is to find two numbers that will give you a product of 24 and a sum of -14.

P: -12*-2 =24
S: -12 + (-2) =14

Therefore your product and sum are -12 and -2.

Next, write these numbers in an expanded quadratic form:
(x-12)(x-2)=0

Now simply solve using null factor law:
12-12=0
2-2=0
Therefore x=12 or 2

Answer 2

Algebraically first then plot the graph

Explanation:

`y= x^2 -14x +24` factorises to `y =(x-12)(x -2)`
So the `x` intercepts are when `y= 0`
This is when `x= 12`. or `x= 2`

The `y` intercept is 24 (when `x` =0)
To find the turning point, use completing the square:

`y = x^2-14x+24`

`=>` `y=(x-7)^2-25` so the turning point is (7,-25)

To do it graphically, it is a `u` shaped parabola that comes down through (0,24) then through (2,0) through the minimum (7,-25) and back up through (12,0)