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

Apr 3, 2018

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 $y = {x}^{2} - 14 {x}^{2} + 24$ is
${x}^{2} - 14 {x}^{2} + 24 = 0$. This makes it easier as it is now a written as a simple quadratic, ready to solve.

So, firstly, the product and sum of $y = {x}^{2} - 14 {x}^{2} + 24$ 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

Apr 3, 2018

Algebraically first then plot the graph

#### Explanation:

$y = {x}^{2} - 14 x + 24$ factorises to $y = \left(x - 12\right) \left(x - 2\right)$
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:

$y = {x}^{2} - 14 x + 24$

$\implies$ $y = {\left(x - 7\right)}^{2} - 25$ 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)