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Quadratics, SMB-015

The diagram shows the curve with equation  `y = x^2-7x + 10`. The curve intersects the `x`-axis at points `A and B`. The point `C` on the curve has the same `y`-coordinate as the `y`-intercept of the curve.
 

 

  1. Find the `x`-coordinates of points `A and B.`  (2 marks)

  2. Write down the coordinates of `C.`  (2 marks)

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  1. `A = 2,\ \ B = 5`
  2. `(7, 10)`
Show Worked Solution
i.    `y` `= x^2-7x + 10`
  `= (x-2) (x-5)`

 
`:.x = 2 or 5`

`:.\ \ x text(-coordinate of)\ \ A = 2`

`x text(-coordinate of)\ \ B = 5`

 

ii.    `y\ text(intercept occurs when)\ \ x = 0`

`=>y text(-intercept) = 10`
 

`C\ text(occurs at intercept:)`

`y` `= x^2-7x + 10` `\ \ \ \ \ text{…  (1)}`
`y` `= 10` `\ \ \ \ \ text{…  (2)}`

 
`(1) = (2)`

`x^2-7x + 10` `= 10`
`x^2-7x` `= 10`
`x (x-7)` `= 10`

 
`x = 0 or 7`

`:.\ C\ \ text(is)\ \ (7, 10)`

Quadratics, SMB-014

The parabola  `y = −2x^2 + 8x`  and the line  `y = 2x`  intersect at the origin and at the point  `A`.
 

Find the  `x`-coordinate of the point `A`.    (2 marks)

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`x=3`

Show Worked Solution
i.  

`x\text(-coordinate of)\ A:`

`y` `= 2x\ \ \ \ \ …\ text{(i)}`
`y` `= -2x^2 + 8x\ \ \ \ \ …\ text{(ii)}`

 
`text(Subst)\ y = 2x\ text{from (i) into (ii):}`

`-2x^2 + 8x` `= 2x`
`-2x^2 + 6x` `= 0`
`-2x (x-3)` `= 0`

  
`:.\ x = 0\ text(or)\ 3`

`:.\ x\text(-coordinate of)\ A\ text(is 3)`