How do you prove that the line y equals 2x plus 5 over 4 touches the curve y squared equals 10x?

Y2=10X and Y=2x+5 will never touch each other. If there is a touch point then there will be a common value (Touch point) of x and y which will satisfy both equations. But there is no common point so it is not possible

Improved Answer:

equation 1: y = 2x++5/4 => y2 = 4x2+5x+1.5625 when both sides are squared

equation 2: y2 = 10x

By definition: 4x2+5x+1.5625 = 10x => 4x2-5x+1.5625 = 0

If the discriminant b2-4ac is equal to zero then the line is tangent to the curve:

b2-4ac = (-5)2-4*4*1.5625 = 0

Therefore the discriminant is zero thus proving that the line is tangent to the curve.