Quadratic Simultaneous Equations (AQA GCSE Further Maths)

Quadratic Simultaneous Equations

What are quadratic simultaneous equations?

• When there are two unknowns (say x and y) in a problem, we need two equations to be able to find them both: these are called simultaneous equations

• If there is an x² or y² or xy in one of the equations then they are quadratic (or non-linear) simultaneous equations

How do I solve quadratic simultaneous equations?

• Use the method of substitution

  • Substitute the linear equation, y = ... (or x = ...), into the quadratic equation

    • Do not try to substitute the quadratic equation into the linear equation

• Solve x² + y² = 25 and y - 2x = 5 

  • Rearrange the linear equation into y = 2x + 5

  • Substitute this into the quadratic equation, replacing all y's with (2x + 5) in brackets

    •  x² + (2x + 5)² = 25

  • Expand and solve this quadratic equation (x = 0 and x = -4)

  • Substitute each value of x into the linear equation, y = 2x + 5, to get their value of y

  • Present your solutions in a way that makes it obvious which x belongs to which y

    • x = 0, y = 5 or x = -4, y = -3

• Check your final solutions satisfy both equations

How do you use graphs to solve quadratic simultaneous equations?

• Plot both equations on the same set of axes

  • to do this, you can use a table of values (or, for straight lines, rearrange into y = mx + c if it helps)

• Find where the lines intersect (cross over)

  • The x and y solutions to the simultaneous equations are the x and y coordinates of the point of intersection

• e.g. to solve y = x² + 3x + 1 and y = 2x + 1 simultaneously, first plot them both (see graph)
  

  • find the points of intersection, (-1, -1) and (0, 1)

  • the solutions are x = -1 and y = -1 or x = 0 and y = 1