Yes you need the distance formula to find the Base and height.
Answer:
![y=(x+12)^2\\](https://tex.z-dn.net/?f=y%3D%28x%2B12%29%5E2%5C%5C)
Step-by-step explanation:
To write a quadratic equation into binomial form we can compare the equation into the completing square form of a quadratic equation like this ,
![y=a(x-h)^2+k](https://tex.z-dn.net/?f=y%3Da%28x-h%29%5E2%2Bk)
now since,
![y=x^2+24x+144](https://tex.z-dn.net/?f=y%3Dx%5E2%2B24x%2B144)
we can equate both the equations from left hand side to right hand side like this,
![x^2+24x+144=a(x-h)^2+k\\](https://tex.z-dn.net/?f=x%5E2%2B24x%2B144%3Da%28x-h%29%5E2%2Bk%5C%5C)
now we solve,
![x^2+24x+144=a(x-h)^2+k\\x^2+24x+144=a((x)^2-2(x)(h)+(h)^2)+k\\x^2+24x+144=a(x^2-2hx+h^2)+k\\x^2+24x+144=ax^2-2ahx+ah^2+k\\](https://tex.z-dn.net/?f=x%5E2%2B24x%2B144%3Da%28x-h%29%5E2%2Bk%5C%5Cx%5E2%2B24x%2B144%3Da%28%28x%29%5E2-2%28x%29%28h%29%2B%28h%29%5E2%29%2Bk%5C%5Cx%5E2%2B24x%2B144%3Da%28x%5E2-2hx%2Bh%5E2%29%2Bk%5C%5Cx%5E2%2B24x%2B144%3Dax%5E2-2ahx%2Bah%5E2%2Bk%5C%5C)
now we compare the coefficients of x^2:
![1 = a\\](https://tex.z-dn.net/?f=1%20%3D%20a%5C%5C)
now we compare the coefficients of x :
![24=-2ah\\24=-2(1)h\\24=-2h\\\frac{24}{-2}=h\\-12=h\\](https://tex.z-dn.net/?f=24%3D-2ah%5C%5C24%3D-2%281%29h%5C%5C24%3D-2h%5C%5C%5Cfrac%7B24%7D%7B-2%7D%3Dh%5C%5C-12%3Dh%5C%5C)
now we compare the constants , (constants are the letters which are not associated with any variable in this case the variable is x)
![144 = ah^2+k\\144=(1)(-12)^2+k\\144=(1)(144)+k\\144=144+k\\144-144=k\\0=k\\](https://tex.z-dn.net/?f=144%20%3D%20ah%5E2%2Bk%5C%5C144%3D%281%29%28-12%29%5E2%2Bk%5C%5C144%3D%281%29%28144%29%2Bk%5C%5C144%3D144%2Bk%5C%5C144-144%3Dk%5C%5C0%3Dk%5C%5C)
so now the value we got all the values for the completing square form we plug those in , a = 1 , h = -12 , k = 0 ,
![y=a(x-h)^2+k\\y=1(x-(-12))^2+0\\y=(x+12)^2\\](https://tex.z-dn.net/?f=y%3Da%28x-h%29%5E2%2Bk%5C%5Cy%3D1%28x-%28-12%29%29%5E2%2B0%5C%5Cy%3D%28x%2B12%29%5E2%5C%5C)
this is the square of a binomial, if you want to verify if we expands this formula by the formula of (a + b)^2 we would get the same result. Thus this is the correct answer.
<span>Let n = number
n = 32 + 8
n = 40</span>