(adsbygoogle = window.adsbygoogle || []).push({}); [SOLVED] trouble finding curve intersection

1. The problem statement, all variables and given/known data

Sketch the area of the region bounded by the curves [tex]y^2+4x=0[/tex] and [tex]y=2x+4[/tex]. Set up two integrals, one with respect to x and one with respect to y, for finding the area of the region. Evaluate one of the integrals to find the area.

2. Relevant equations

3. The attempt at a solution

First I rearrange the equations of the curves to isolate both x & y for each.

[tex]y^2+4x=0 \Longrightarrow y^2=-4x \Longrightarrow \sqrt{y^2}=\sqrt{-4x} \Longrightarrow y=2\sqrt{-x},-2\sqrt{-x}[/tex]

[tex]y^2+4x=0 \Longrightarrow x=-\frac{1}{4}y^2[/tex]

[tex]y=2x+4[/tex]

[tex]y=2x+4 \Longrightarrow x=\frac{y-4}{2} \Longrightarrow x=\frac{1}{2}y-2[/tex]

Next I must find the domain of the integrals by setting y=y.

[tex]-\frac{1}{4}y^2=\frac{1}{2}y-2 \Longrightarrow \frac{1}{4}y^2+\frac{1}{2}y-2 \Longrightarrow \frac{1}{2}(y-2)(y+4) \Longrightarrow y=2,-4[/tex]

This is where I'm running into trouble, when I set x=x.

[tex]2\sqrt{-x}=2x+4[/tex] [tex]-2\sqrt{-x}=2x+4[/tex]

I know the points of intersection are -1 & -4 respectively but I can't seem to tease those out of the equations.

[tex]2\sqrt{-x}=2x+4 \Longrightarrow 2x^\frac{1}{2}+2x+4 \Longrightarrow 2(x^\frac{1}{2}+x+2)[/tex]

[tex]-2\sqrt{-x}=2x+4 \Longrightarrow -2x^\frac{1}{2}+2x+4 \Longrightarrow 2(-x^\frac{1}{2}+x+2)[/tex]

I'm not sure what to do from here. I can't factor this and if I square it and apply the quadratic equation I get a negative under the sqrt. Some advice would be greatly appreciated. Cheers.

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# Trouble finding curve intersection

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