The graph of the parabola \(x = 2y^2 - 6y + 3\) has an x-intercept \((a,0)\) and two -intercepts \((0,b)\) and (0,c) . Find a+b+c.
x = 2y^2 - 6y + 3
When y = 0 , x = 3 so ( a, 0) means that a = 3
To find the y intercepts, let x =0 and use the quadratic formula on y
y = [ 6 +/- sqrt (36 - 4(2)(3)] / 4 = [ 6 +/- sqrt (12)] / 4 =
[ 6 +/- 2sqrt(3)]/ 4 = [3 +/- sqrt(3)]/2
So....let b = [3 + sqrt (3)]/ 2 and let c = [3 - sqrt (3)] / 2
So.....a + b + c =
3 + [3 + sqrt(3)]/2 + [3 - sqrt (3)] / 2 =
3 + 3/2 + 3/2 =
6
Here's a graph that confirms this : https://www.desmos.com/calculator/izhnkrojmb