Practise the method.
Example 1
A quadratic graph has equation y=x2+bx+c. It passes through (0,33) and (1,28). A straight line passes through (0,5) and (2,15). Find both x−coordinates where the graphs meet.
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Answer: x=4x=7
- The quadratic has c=33 and b=−6.
- The line is y=5x+5.
- Equating gives x2−11x+28=0
- x=4 or 7.
Example 2
Machine A follows a linear rule: input 1 gives 25; input 3 gives 35. Machine B takes A's output, squares it, then subtracts 4. Find both original inputs that produce final output 896.
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Answer: x=2x=−10
- A outputs 5x+20.
- Therefore (5x+20)2=900
- A's output is 30 or −30.
- Solve both linear equations: x=2 or −10.
Example 3
A quadratic graph has equation y=x2+bx+c. It passes through (0,29) and (1,23). A straight line passes through (0,2) and (2,12). Find both x−coordinates where the graphs meet.
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Answer: x=3x=9
- The quadratic has c=29 and b=−7.
- The line is y=5x+2.
- Equating gives x2−12x+27=0
- x=3 or 9.