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Quadratic Simultaneous Equations (Cambridge O Level Additional Maths)
Revision Note
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
- Substitute the linear equation, y = ... (or x = ...), into the quadratic 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
Examiner Tip
- If the resulting quadratic has a repeated root then the line is a tangent to the curve (as there is only 1 solution)
- If the resulting quadratic has no roots then the line does not intersect with the curve (as there are 0 solutions) – or you have made a mistake!
- When giving your final answer, make sure you indicate which x and y values go together
- If you don’t make this clear you can lose marks for an otherwise correct answer
Worked example
Solve the equations
x2 + y2 = 36
x = 2y + 6
Number the equations.
There is one quadratic equation and one linear equation so this must be done by substitution.
Equation (2) is equal to so this can be eliminated by substituting it into the part for equation (1).
Substitute into equation (1).
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Expand the brackets, remember that a bracket squared should be treated the same as double brackets.
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Simplify.
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Rearrange to form a quadratic equation that is equal to zero.
The question does not give a specified degree of accuracy, so this can be factorised.
Take out the common factor of .
Solve to find the values of .
Let each factor be equal to 0 and solve.
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Substitute the values of into one of the equations (the linear equation is easier) to find the values of .
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