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# Algebra

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The parabolas defined by the equations y = x^2 + 4x + 6 and y = 1/2*x^2 + x + 4 intersect at (a,b) and (c,d) where c >= a. What is c - a?

May 17, 2022

#1
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Where c >= a. What is c - a?

Hello Guest!

$$y = x^2 + 4x + 6$$

$$y = 1/2*x^2 + x + 4$$

(a,b),(c,d)

$$x^2 + 4x + 6 =\frac{1}{2}x^2 + x + 4\\ \frac{1}{2}x^2+3x+2=0\\ x^2+6x+4=0\\ x=-3\pm \sqrt{9-4}\\ x\in \color{blue}\{(-\sqrt{5}-3),(\sqrt{5}-3)\}$$

a                   c

$$b = (-\sqrt{5}-3)^2 + 4(-\sqrt{5}-3) + 6 \\ \color{blue}b=12.472\\ d=(\sqrt{5}-3)^2 + 4(\sqrt{5}-3) + 6 \\ \color{blue}d=3.528$$

c > a

$$c-a=(\sqrt{5}-3)-(-\sqrt{5}-3)\\ \color{blue}c-a=2\sqrt{5}$$

!

May 17, 2022