How is the distance to the x-axis related to the roots of quadratic equations?

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SUMMARY

The discussion centers on the relationship between the roots of quadratic equations and their distance from the x-axis. Specifically, the quadratic equation in standard form, ax² + bx + c = 0, has roots determined by the formula x = (-b ± √(b² - 4ac))/2a. The distance from the vertex of the parabola to the x-axis is calculated as c - (4b²/a), which is relevant when x = b/2a. This distance is crucial for understanding the behavior of the quadratic function, particularly when analyzing its minimum or maximum points.

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lmamaths
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Hi,

How is the roots of a quadratic equation related
to the distance from the x-axis at where
the root is -
where ...
ax^2+bx+c=0
and ...
x = (-b +- SQRT(b^2-4ac))/2

Can someone help me to establish where this
distance relationship to the x-axis and the root
come from?

Thx!
LMA
 
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Am I understanding this correctly? Are you asking how the roots of a quadratic equation relate to "the distance from the x-axis at where the root is"? A root of an equation, by definition, is a point where y= 0. if y=0, then the distance from the x-axis is 0: the graph crosses the axis there!
 
Hi,

Maybe I mean't lowest part of the curve to the x-axis, consider:
y=2x^2-3x+2

Thx!
Leo
 
lmamaths said:
Hi,

Maybe I mean't lowest part of the curve to the x-axis, consider:
y=2x^2-3x+2

Thx!
Leo

I'm not really sure what you're asking Leo. That function has no real roots. If you extend the domain to complex numbers then the function is still zero (both real and imaginary parts) at each of its complex zeros. Real or complex the function is still zero at it's zero's.
 
lmamaths said:
Hi,

Maybe I mean't lowest part of the curve to the x-axis, consider:
y=2x^2-3x+2

Thx!
Leo


Ah- distance from the vertex to the x-axis.

Given y= ax2+ bx+ c= 0, consider solving by completing the square: write this as a(x2+ (b/a)x+ b2/4a2)+ c- 4b2/a= a(x- (b/2a))2+ c- 4b2/a.

The distance from the x-axis to the vertex is given when x = b/2a and is
c- 4b2/a. do you see how that is connected to the value of x that satisfies the equation?
 

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