Determine the quadratic function with given roots and a given point. Express each function in standard form. a) x=2 +or- √5, (2,10)  b) x=1/3 and x=-4, (1,5)

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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1) Since the problem provides the roots of quadratic equation, hence you need to remember that you may use Lagrange's resolvents such that:

`(x - x_1)(x - x_2) = x^2 + qx + p`

You need to substitute `2+-sqrt5 ` for `x_(1,2)`  such that:

`(x - 2 - sqrt5)(x - 2 + sqrt5) = (x - 2)^2 - (sqrt 5)^2`

`(x - 2 - sqrt5)(x - 2 + sqrt5) = x^2 - 4x + 4 - 5`

`(x - 2 - sqrt5)(x - 2 + sqrt5) = x^2 - 4x - 1`

Since the problem provides the roots, there is no need to give a point also. Notice that the given point (2,10) does not belong to the parabola given by quadratic equation `f(x) = x^2 - 4x - 1` .

2) Since the problem provides the roots of quadratic equation, hence you need to remember that you may use Lagrange's resolvents such that:

`(x - 1/3)(x + 4) = x^2 + 4x - x/3 - 4/3`

`(x - 1/3)(x + 4) = x^2 + (11/3)x - 4/3`

Hence, evaluating the quadratic function under the given conditions yields `f(x) = x^2 + (11/3)x - 4/3`  (notice that the point (1,5) does not belong to the graph of the determined function).

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