Graphical polynomial solutions
The solutions of P(x)=Q(x) are the horizontal coordinates of the intersections of the two graphs. Use either rule with an unrounded x-value to obtain the matching vertical coordinate. Use the equation to verify exact values or justify finer rounding
- Set the outputs equal At an intersection, both rules have the same output for the same input \(\displaystyle P(x)=Q(x)\Longleftrightarrow P(x)-Q(x)=0\)
- Use horizontal coordinates as solutions Each intersection in the stated window gives one solution through its x-coordinate \(\displaystyle P(a)=Q(a)\Rightarrow x=a\text{ is a solution}\)
- Recover the vertical coordinate Substitute the unrounded intersection x-coordinate into either original rule. Retain guard digits and round each requested coordinate only at the end \(\displaystyle x=a\Rightarrow y=P(a)=Q(a)\)
- Match the graph precision Use the graph to locate intersections. Verify exact values and resolve uncertain rounding with the displayed rules or a suitable numerical method \(\displaystyle \text{one-unit grid}\not\Rightarrow\text{two decimal places}\)
Worked example
Find the intersection points of the two polynomial rules below
- \(\displaystyle P(x)=Q(x)\Rightarrow x^2-3x+5=x+2\Rightarrow x^2-4x+3=0\)Equate the two outputs and move all terms to one side
- \(\displaystyle x^2-4x+3=\left(x-1\right)\left(x-3\right)=0\Rightarrow x=1\text{ or }x=3\)Factor the difference and use the null factor law to find the horizontal coordinates
- \(\displaystyle Q(1)=1+2=3\Rightarrow\left(1,3\right),\qquad Q(3)=3+2=5\Rightarrow\left(3,5\right)\)Substitute each solution into Q to obtain the corresponding vertical coordinate
- \(\displaystyle P(1)=3=Q(1),\qquad P(3)=5=Q(3)\)The equal values verify both intersection points in the original rules
Common mistakes
Difficulty levels
Questions about this topic
Why does an intersection solve P(x)=Q(x)?+
At an intersection both graphs have the same x-coordinate and the same y-coordinate, so their outputs are equal for that input
How do you find the y-coordinate after reading an intersection?+
Substitute the unrounded x-coordinate, or retain sufficient guard digits, in either original rule. Round y only at the end; using an already rounded x can change its rounded value
Why can a graph give only an approximate solution?+
The grid and viewing window limit how precisely a coordinate can be read. Use the equation to justify finer precision. A sign-changing bracket can refine a crossing, but an even-multiplicity contact needs a repeated-root solver or separate exact verification
Can I use a calculator?+
A calculator is allowed unless a question is marked no calculator
Which year level and chapter is this?+
Year 11, VCE Units 1 & 2, Mathematical Methods, chapter 5: Higher-degree polynomials