- Solutions are always read off the
*x*-axis - Solutions of
**f(**are where the graph of*x*) = 0**y = f(**crosses the*x*)*x*-axis - If asked to use the graph of
**y = f(x)**to solve a**different**equation (the question will say something like “by drawing a suitable straight line”) then:- Rearrange the equation to be solved into
**f(**and draw the line*x*) = mx + c**y = mx + c** - Solutions are the
-coordinates of where the**x****line**(y = mx + c) crosses the**curve**(y = f(x)) -
E.g. if given the curve for y = x

^{3}+ 2x^{2 }+ 1 and asked to solve x^{3}+ 2x^{2 }− x − 1 = 0, then;- rearrange x
^{3}+ 2x^{2 }− x − 1 = 0 to x^{3}+ 2x^{2 }+ 1 = x + 2 - draw the line y = x + 2 on the curve y = x
^{3}+ 2x^{2 }+ 1 - read the x-values of where the line and the curve cross (in this case there would be 3 solutions,
**approximately**x = -2.2, x = -0.6 and x = 0.8);

- rearrange x

- Rearrange the equation to be solved into

- Note that
**solutions**may also be called**roots**

**Plot**both equations on the same set of axes using**straight line graphs y = mx + c**- Find where the lines
**intersect**(cross)- The
*x*and*y***solutions**to the simultaneous equations are the*x*and*y***coordinates**of the point of**intersection**

- The
- e.g. to solve 2
*x*-*y*= 3 and 3*x*+*y*= 4 simultaneously, first plot them both (see graph)

- find the point of intersection, (2, 1)
- the solution is
*x*= 2 and*y*= 1

- e.g. to solve
*y*=*x*^{2}+ 4*x*− 12 and*y*= 1 simultaneously, first plot them both (see graph)

- find the two points of intersection (by reading off your scale), (-6.1 , 1) and (-2.1, 1) to 1 decimal place
- the solutions from the
**graph**are**approximately***x*= -6.1**and***y*= 1 and*x*= 2.1**and***y*= 1- note their are
**two pairs of***x*,*y*solutions - to find
**exact**solutions, use**algebra**

- note their are

- If solving
**an equation**, give theonly as your final answer*x*values - If solving a pair of
**linear****simultaneous equations**give anas your final answer*x*and a*y*value - If solving a pair of simultaneous equations where
**one is linear and one is quadratic,**give two**pairs of**as your final answer*x*and*y*values

The graph of is shown below.

Use the graph to estimate the solutions of the equation . Give your answers to 1 decimal place.

We are given a different equation to the one plotted so we must rearrange it to (where is the plotted graph)

Now plot on the graph- this is the solid red line on the graph below

The solutions are the coordinates of where the curve and the straight line cross so

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