Answer:
[tex]r = \±\sqrt{14[/tex]
[tex]Product = -14[/tex]
Step-by-step explanation:
Given
[tex]\frac{1}{2x} = \frac{r - x}{7}[/tex]
Required
Find all product of real values that satisfy the equation
[tex]\frac{1}{2x} = \frac{r - x}{7}[/tex]
Cross multiply:
[tex]2x(r - x) = 7 * 1[/tex]
[tex]2xr - 2x^2 = 7[/tex]
Subtract 7 from both sides
[tex]2xr - 2x^2 -7= 7 -7[/tex]
[tex]2xr - 2x^2 -7= 0[/tex]
Reorder
[tex]- 2x^2+ 2xr -7= 0[/tex]
Multiply through by -1
[tex]2x^2 - 2xr +7= 0[/tex]
The above represents a quadratic equation and as such could take either of the following conditions.
(1) No real roots:
This possibility does not apply in this case as such, would not be considered.
(2) One real root
This is true if
[tex]b^2 - 4ac = 0[/tex]
For a quadratic equation
[tex]ax^2 + bx + c = 0[/tex]
By comparison with [tex]2x^2 - 2xr +7= 0[/tex]
[tex]a = 2[/tex]
[tex]b = -2r[/tex]
[tex]c =7[/tex]
Substitute these values in [tex]b^2 - 4ac = 0[/tex]
[tex](-2r)^2 - 4 * 2 * 7 = 0[/tex]
[tex]4r^2 - 56 = 0[/tex]
Add 56 to both sides
[tex]4r^2 - 56 + 56= 0 + 56[/tex]
[tex]4r^2 = 56[/tex]
Divide through by 4
[tex]r^2 = 14[/tex]
Take square roots
[tex]\sqrt{r^2} = \±\sqrt{14[/tex]
[tex]r = \±\sqrt{14[/tex]
Hence, the possible values of r are:
[tex]\sqrt{14[/tex] or [tex]-\sqrt{14[/tex]
and the product is:
[tex]Product = \sqrt{14} * -\sqrt{14}[/tex]
[tex]Product = -14[/tex]
