Quadratics - minimum turning point :)

hi !! :) thank you so much in advance !! :) i am revising for my standard grade maths came across these and my mind went blank !! panicking now !!

i have a diagram and it says find the minimum turning point
the equation is x²-2x-15
i think i have already found the roots ? but im not sure can you check if they are 5 and -3

thank you sooooo much !! :D

~ stephanie lyra skye !! :)

Yes, your roots are correct. That means that the curve intercepts the X-axis at X=5 and -3.

Since the equation is +x^2, that means each end of the curve points upwards. So the minimum turning point should be the midpoint of the two x-intercepts.

Midpoint of x=5 and -3 is X=1
Putting x=1 into the equation x^2-2x-15 gives you the full coordinates (1,-16)

x^2-2x-15
=(x+3)(x-5)
x=5.-3

This value of x is crossing points of x-axis and the equation of x^2-2x-15.

To get to know the minimum turning point,
it's good to use completing square.

x^2-2x-15
↑using x^2and-2x makes (x+O)^2

=x^2-2x+1-1-15
=(x-1)^2-16


co ordinate(1,-16) is the minimum turning point the equation of x^2-2x-15



If you want to know crossing points of x-axis and an equation,
you can get the value of x by doing factorization
If you want to know the minimum turning point,
you can get the value of x,y by doing completing square.

make y equal 0 to find x
0=2x²-12x
take out common factors
0=2x (x - 6)
according to the null factor law (if 2 factors times together and equal 0 than one of the factors has to equal 0) we need to make each factor = 0 in separate steps, to find what the other factor would equal. The factors that times together in this equation are (2x) and (x-6))
first I'll make (x- 6) = 0
0 = 2x
x=0
now make 2x =0
0=x-6
x=6
so your roots are 0 and 6