algebra – quadratics - Show that the x-coordinates of the points of intersection of y = x (4 – x) and y = x^2 (7 - x) are given by the solutions to the equation x(x^2 - 8x + 4). Find the exact coordinates of intersection point?

Krishna (last edited 4 years ago) (Expert, Educator @Qalaxia)

Step 1: Solve two equations(linear equations in two variables) to find the coordinate (x or y) at the point of intersection

NOTE: If line intersecting the y-axis then put x = 0 in the equation or

If line intersecting the x-axis then put y = 0 in the equation.

EXAMPLE: y = 2x + 3, y = -0.5x + 7

Setting the two y-coordinates equal looks like this:

2x + 3 = -0.5x + 7

2.5x + 3 = 7

2.5x - 4 = 0

Step 2: Simplify the equation to find the unknown value.

NOTE: Apply inverse operations on both sides, for linear equations.

Apply the any one method to solve the quadratic equation.

EXAMPLE: 9 + 8k + 8 <0

8k+17<0

8k < -17

k < \frac{-17}{8}

Step 3: Substitute the coordinator (calculated in step 1) in the either of the equations to find the another coordinate. (x or y)

NOTE: The y-coordinate can be found by placing the x-coordinate, into either of the equations for the lines and solving for y vise versa.

Step 4: You now have the x-coordinate and y-coordinate for the point of intersection.

NOTE: As a check for your work plug the x-coordinate into the other equation and you should get the same y-coordinate.