express 2x² - 12x + 23 in the form a(x+b)² +c?

Question

how would i sketch the curve y=2x² - 12x + 23
how would i find the points

plzz help mee...im confused!

tanx

Answer 1

Factor 2 out of the terms involving x

2(x^2 - 6x) + 23

Add and subtract the square of half of 6. (9)

2(x^2 - 6x + 9 - 9) + 23

take the subtracted 9 out of the brackets, remembering that it is multiplied by 2.

2(x^2 - 6x + 9) + 23 - 2(9)

Factor the trinomial in brackets. You can do this easily because it is a prefect square.

2(x-3)(x-3) + 23 - 2(9)

Simplify

2(x-3)^2 + 5

Answer 2

express 2x² - 12x + 23 in the form a(x+b)² +c?

2x² - 12x =2(x^2 -6x)

let's make x^2-6x as a square of (x-3)
(x-3)^2 = x^2 -6x +9

so x^2-6x can be written as (x-3)^2 -9

2x² - 12x = 2(x-3)^2 -18

2x² - 12x + 23 = 2(x-3)^2 -18 +23 =
2(x-3)^2 +5

hence a =2; b =-3 and c =5

Answer 3

2 x^2 - 12 x + 23
= 2 [ x^2 - 6 x + 23/2 ]
= 2 [ x^2 - 6 x + 3 ^2 - 3^2+ 23/2 ]
= 2 [ (x - 3)^2 - 9 + 23/2 ]
= 2 [ (x - 3)^2 - 18/2 + 23/2 ]
= 2 [ (x - 3)^2 + 25/2 ]
= 2 (x - 3)^2 + 25

Answer 4

2(x² - 6x + 23/2)
2(x² - 6x + 9 - 9 + 23/2)
2(x² - 6x + 9) - 18 + 23
2(x - 3)² + 5
y = 2(x - 3)² + 5

Points on curve
(3 , 5) , (0 , 23) , (4 , 7)

(3,5) is a minimum turning point.

line x = 3 is an axis of symmetry
cuts y axis at y = 23
Does not cut x axis.

Answer 5

2x^2 - 12x + 23

take 2 as factor out

2(x^2 - 6x + 23/2)

complete the square

=>2[(x^2 - 2(3) x + 3^2 - 3^2 +23/2)]

=>2[(x - 3)^2 - 9 + 23/2)]

=>2[(x-3)^2 +5/2]

=>2(x-3)^2 + 5

Answer 6

put in the form (ax+b)^2+c=a^2x^2+2abx+(b^2+c)
equate coefficients of x > 2x^2- 12x + 23
Then a^2=2, or a=rt(2), 2ab=-12 or b=-3rt(2), and b^2+c=23 or c=5.
>(rt(2)x-3rt(2))^2+5 or 2(x-3)^2+5
Curve has a min TV at (3,5), it's a parabola, plot it on a grafix calculator, or on a grid and put in values of x, say 1,2,3..etc and calculate y-values in 2(x-3)^2+5 for example.
Getting deja-vu here?! There is a formula for any quadratic you know...!

Answer 7

y = 2(x^2 - 6x + 11.5)
y = 2(x-3)^2 + 5

Answer 8

a generic way of converting would be

for mx^2 + nx + p = m(x + n/2m)^2 + (c-n^2/4m^2)