Question 1209249
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Find the smallest value of x such that x^2 + 10x + 25 = -x^2 + 5x + 32.
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                                        Step by step


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              <U>Step 1.  Reduce to the standard form quadratic equation</U>


For it, collect all the terms in the left side and combine like terms.
You will get

    2x^2 + 5x - 7 = 0.



              <U>Step 2.  Calculate the discriminant to determine if the roots are real</U>


d = b^2 - 4ac = 5^2 - 4*2*(-7) = 25 + 56 = 81.


The discriminant is positive, so the roots are real numbers, 
and we can use the conception "greater-smaller" for them.



              <U>Step 3.  Use the quadratic formula to find the smallest root</U>


smallest root = {{{(-b - sqrt(d))/(2a)}}} = {{{(-5 - sqrt(81))/(2*2)}}} = {{{(-5 - 9)/4}}} = {{{(-14)/4)}}} = -3{{{1/2}}} = -3.5.    <U>ANSWER</U>  
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Solved.