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

In summary, the conversation discusses the relationship between the roots of a quadratic equation and the distance from the x-axis at where the root is located. The distance from the vertex to the x-axis is given by c-4b^2/a when x = b/2a. This is connected to the value of x that satisfies the equation.
  • #1
lmamaths
6
0
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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  • #2
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!
 
  • #3
Hi,

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

Thx!
Leo
 
  • #4
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.
 
  • #5
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?
 

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

The distance to the x-axis is directly related to the roots of quadratic equations. This is because the roots of a quadratic equation represent the points where the graph of the equation intersects with the x-axis. Therefore, the distance from the x-axis to the roots is the same as the distance from the x-axis to the points of intersection.

2. Is there a specific formula for calculating the distance to the x-axis from the roots of a quadratic equation?

Yes, there is a formula for calculating the distance from the x-axis to the roots of a quadratic equation. It is given by the formula d = (-b ± √(b²-4ac)) / 2a, where a, b, and c are the coefficients of the quadratic equation in the form ax²+bx+c=0.

3. How can the distance to the x-axis help in solving quadratic equations?

The distance to the x-axis can help in solving quadratic equations by providing important information about the nature of the roots. If the distance is positive, it means that the roots are real and distinct. If the distance is zero, it means that the roots are real and equal. And if the distance is negative, it means that the roots are complex conjugates.

4. Can the distance to the x-axis ever be negative?

Yes, the distance to the x-axis can be negative in certain cases. This occurs when the discriminant (b²-4ac) of the quadratic equation is negative, resulting in complex roots. In this case, the distance to the x-axis represents the imaginary part of the complex roots.

5. Is the distance to the x-axis always the same for all quadratic equations?

No, the distance to the x-axis can vary for different quadratic equations. It depends on the values of the coefficients a, b, and c, which can differ from one equation to another. However, for a specific quadratic equation, the distance to the x-axis remains constant regardless of the value of x.

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