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## Homework Statement

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Find equations of both lines through the point (2,-3) that are tangent to the parabola y=x

^{2}+x.

## Homework Equations

Slope formula: (Y

_{2}-Y

_{1})/(X

_{2}-X

_{1}) = M

## The Attempt at a Solution

Here's what I think I need to do. First I think I need to find the derivative of the parabola. Then I need to use the point (2,-3) as point 2 in the slope equation Y

_{2}-Y

_{1}/X

_{2}-X

_{1}to find two X values that give me a slope equal to the derivative at that x-value.

Derivative: y'=2x+1

Finding slope: The tangent line passes through (2,-3) and the parabola. Since the slope of the tangent line is found by the derivative, that means that y

_{2}-y

_{1}/x

_{2}-x

_{1}should be equal to the derivative 2x+1, or: [-3-(2x+1)]/(2-x) = 2x+1

So that becomes: (-3-2x-1)/(2-x) = 2x+1, then (-4-2x)/(2-x) = 2x+1.

Moving the bottom over: -4-2x = (2x+1)(2-x)

Foiling: -4-2x = 4x-2x

^{2}+2-x.

Combining: -4-2x = -2x

^{2}+3x+2

Moving the right side over and combining: 2x

^{2}-5x-6 = 0

So the tangent lines should be located at the x-coordinates found by finding x in the above quadratic, right?