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The problem provides the information that the graph touches x axes one time, hence, the quadratic equation has two equal roots.
You should remember the case a quadratic equation `ax^2 + bx + c = 0` has two equal solutions such that:
`Delta = 0 =gt b^2 - 4ac = 0`
Comparing the given equation to standard form of quadratic yields `a = 1 , b = -(m-3) , c = m` .
Substituting these values in formula of `Delta` yields:
`Delta = (3-m)^2 - 4m`
You should expand the square such that:
`9 - 6m + m^2 - 4m = 0`
`m^2 - 10m + 9 = 0`
You should use quadratic formula to find m such that:
`m_(1,2) = (10+-sqrt(100-36))/2`
`m_(1,2) = (10+-sqrt64)/2`
`m_(1,2) = (10+-8)/2`
`m_1 = 9 ; m_2 = 1`
Hence, evaluating the possible values for m that follows the given conditions yields `m_1 = 9` and `m_2 = 1` .
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