How is discriminant related to real x?Solve for range of the function, y = x 2 + 4 x − 1 3 x 2 + 12 x + 20 Text book says, cross multiply and express the obtained equation as a quadratic equation in xSo I get ( 3 y − 1 ) x 2 + ( 12 y − 4 ) x + ( 20 y − 1 ) = 0Now it says find discriminant D, so we have, D = − 4 ( 3 y − 1 ) ( 8 y + 3 )Now it says, set D ≥ 0 as x is real. Wait what?Isn't x ∈ R the domain for a quadratic function? Meaning "x" is always real? What does a discriminant got anything to do with x being real, when all discriminant tells us is whether or not the ROOTS are real? Help please.To be more specific about my doubt, here's an edit.

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How is discriminant related to real x?Solve for range of the function,  y  =                    x        2            +      4      x      −      1              3              x        2            +      12      x      +      20      Text book says, cross multiply and express the obtained equation as a quadratic equation in xSo I get   (  3  y  −  1  )      x    2    +  (  12  y  −  4  )  x  +  (  20  y  −  1  )  =  0Now it says find discriminant D, so we have,  D  =  −  4  (  3  y  −  1  )  (  8  y  +  3  )Now it says, set   D  ≥  0 as x is real. Wait what?Isn&#039;t   x  ∈      R   the domain for a quadratic function? Meaning &quot;x&quot; is always real? What does a discriminant got anything to do with x being real, when all discriminant tells us is whether or not the ROOTS are real? Help please.To be more specific about my doubt, here&#039;s an edit.

Phiplyrhypelw0 · Accepted Answer

Initial Volume = 20l 20% juiceLet after &#039;x&#039; l of 20% juice to removed. We are adding 70% juice of &#039;x&#039; litreoTotal concetration of juice to 40%Total volume is again 20 l.  therefor  (  20  −  x  ) has 20% juicex has 70% juicemixture has 40% juice            (      20      −      x      )      20      %      +      x      ×      70      %        20    ×  100  =  40    (            20      −      x        5    +            7      x        10    )  ×  5  =  40              40      −      2      x      +      7      x        10    =  8    40  +  5  x  =  80    5  x  =  +  40    x  =  +  8It means, after removing 8l of 20% juice.we should add 70% juice to get 40% juice

Hayley Mcclain · Accepted Answer

Step 1                    x        2            +      4      x      −      1              3              x        2            +      12      x      +      20        =  yAfter cross multiplying, we obtain  (  3  y  −  1  )      x    2    +  (  12  y  −  4  )  x  +  (  20  y  +  1  )  =  0Step 2If the discriminant is nonnegative, then we can find real x as solution to the quadratic equation, and hence given a particular value of y that makes the discriminant nonnegative, we can find x that satisfy the original equation.However, if the discriminant is negative, then x that satisfies the quadratic equation is not real.

How is discriminant related to real x? Solve for range of the function, y=(x^2+4x-1)/(3x^2+12x+20)

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