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**Coursework Notes - Algebra**

**Graphical Solutions**

vertical line & quadratic curve |
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A 'straight line intersecting a straight line' is dealt within 'simultaneous equations' here

__Vertical line intersecting a quadratic curve__

__Example__ Find the point of intersection when the vertical at x=-2 meets the curve,

Substitute the value of x=-2 into the quadratic equation to find y.

__hence the point of intersection is (-2, -3)__

__Horizontal line intersecting a quadratic curve__

__Example__ Find the two points of intersection when the horizontal at y=4 meets the curve,

To find the two points, put one equation equal to the other, rearrange putting zero on one side and find the roots.

The roots are complex, therefore we use the quadratic equation formula:

__The two points of intersection are (1.828, 4) and (-3.828, 4)__

N.B. the rounding of square roots makes the answers only approximate

__Angled straight line intersecting a quadratic curve __

__Example__ - Find the points of intersection when the straight line with equation,

meets the curve,

As with the horizontal line intersection , the solution is to put one equation equal to the other, rearrange, put zero on one side and find the roots.

__The two points of intersection are(0.76, -0.93) and (-1.99, 2.99)__

__Straight line intersecting a circle __

__Example__ - Find the points of intersection when the straight line with equation,

meets the circle with equation,

The solution is to take the y-value from the straight line equation and put it into the y-value of the circle equation. Then solve for x.

__The two points of intersection are(2.68, 1.34) and (-2.68, -1.34)__

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