Mammoth Memory

Substituting values of x

For many quadratic equations they can actually be solved by substituting values of x into the equation and then drawing the graph.

If you have time our advice is:

Always do a quick sketch 

Example 1

Solve `x^2+3x-4=y`

 

`x=2` `2^2+3times2-4` `=` `4+6-4` `=6`
`x=1` `1^2+3times1-4` `=` `1+3-4` `=0`
`x=0` `0^2+3times0-4` `=` `0+0-4` `=-4`
`x=-1` `(-1)^2+3times(-1)-4` `=` `1-3-4` `=-6`
`x=-2` `(-2)^2+3times(-2)-4` `=` `4-6-4` `=-6`
`x=-3` `(-3)^2+3times(-3)-4` `=` `9-9-4` `=-4`
`x=-4` `(-4)^2+3times(-4)-4` `=` `16-12-4` `=0`
`x=-5` `(-5)^2+3times(-5)-4` `=` `25-15-4` `=6`

 

If we plot these coordinates on a graph we get:

Plot the coordinates on a graph of the broken down equation

Now joining up the points we get:

Join the dots through the x axis then through the y showing the roots and a parabola

Here we can see that the roots are `+1` and `-4`

 

So why do we not use this way always?

Because you might get a beast like:

`2x^2-5x-6=y`

Try plotting this one:

`x=4` `2times4^2-5times4-6` `=` `32-20-6` `=6`
`x=3` `2times3^2-5times3-6` `=` `18-15-6` `=-3`
`x=2` `2times2^2-5times2-6` `=` `8-10-6` `=-8`
`x=1` `2times1^2-5times1-6` `=` `2-5-6` `=-9`
`x=0` `2times0^2-5times0-6` `=` `0-0-6` `=-6`
`x=-1` `2times(-1)^2-5times(-1)-6` `=` `2+10-6` `=6`

 

If we plot these coordinates on a graph we get:

Plot the coordinates on a graph of the broken down equation

Now joining up the points we get:

Join the dots through the x axis then through the y showing the roots and a parabola

You can see that you can not find the roots graphically.

But you can find them using the quadratic formula.

 

Example 2

Solve `2x^2-5x-6=y`  using the quadratic formula.

The quadratic formula is:

`x=(-b+-sqrt(b\ ^2-4ac))/(2a)`

In this case `a=2`, `b=-5`  and `c=-6`

`x=(-(-5)+-sqrt((-5)^2-4times2times(-6)))/(2times2)`

`x=(5+-sqrt(25+48))/4`

`x=(5+-sqrt73)/4`

`x=(5+sqrt73)/4 \ \ \ or \ \ \ x=(5-sqrt73)/4`

`x=3.386 \ \ \ or \ \ \ x=-0.886`

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