Why do we set each factor equal to 0 and then solve for x wh

Why do we set each factor equal to 0 and then solve for x when solving a quadratic equation? We define the slope of a line as delta y/delta x or y_2 - y_1/x_2 - x_1. But when we have a function of the form y = ax (where b = 0), we can define the slope as simply y/x. Why is this? If f(x) = ax + b, where a ne 0, what is the difference between the x-intercepts, the zeros, and the roots of a function? Your study buddy says that if y^2 = x^2 + a^2, then y = x + a. Is this correct? If so, explain. If not, find values for x and a to show that it is not correct. Why is the slope of a vertical line considered to be \"undefined\"? What is the slope of a horizontal line? Can a parabola have one real root and one imaginary root? Explain. What does the slope of a secant line drawn between two points on a function represent? Explain your answer. Once you use a formula to find the vertex of a parabola, how can you determine if it is a maximum or minimum value? When simplifying rational expressions with complex numbers in the denominator, why do we multiply the top and bottom of the given fraction by the conjugate of the denominator? How do you find the x-intercepts and y-intercepts of a graph of an equation? What is the equation of the x-axis? What is the equation of the y-axis? Why? Why does the \"vertical line test\" help us to identify functions versus non-functions? What is a rational function?, how can you determine the vertical and horizontal asymptotes of rational functions?

Solution

1)
Say we had x^2 - 3x + 2 = 0
When we factor, we get (x - 2)(x- 1) = 0

Now, either one of those factors can be 0 resulting in a combined
product of ZERO.

This is why we have to set x- 2 = 0 and x - 1 = 0
to get the two zeros are 1 and 2

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2)
y = ax has a point (0,0) on it

So one point is already (x1,y1) = (0,0)

So, (y2-y1)/(x2-x1) is the same as :
(y2 - 0) / (x2 - 0)
or y2 / x2

Or simply y/x

This is why

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3)
f(x) = ax + b

x-int :
Plug in y or f(x) as 0 and solve for x
ax + b = 0
x = -b/a
So, the x-int is (-b/a , 0)

zeros and roots are just fancy terms for x-intercepts

The x-intercept is the point (-b/a , 0) on the graph of ax + b
The zero is simply the value at which the graph crosses the x-axis
Root is same as zero

All the values, x-int , root and zero for this function are -b/a
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4)
y^2 = x^2 + a^2

When we square root both sides, we get
y = sqrt(x^2 + a^2), which by no way is x + a

Example.... We can choose x = 5, a = 12 and y = 13

Notice y^2 = x^2 + a^2
13^2= 12^2 + 5^2
169 = 169
is TRUE

Does y = x + a?
13 = 12 + 5
13 = 17
NOPE!

So, it aint true and the values of x and a that shows it is :
x = 5 and a = 12