SOLUTION: algebra of linear and quadratic expressions. find the values of k for which the quadratic 2x^2+kx+9=0 has one real solution, and another with no real solutions.

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Question 155037: algebra of linear and quadratic expressions.
find the values of k for which the quadratic 2x^2+kx+9=0 has one real solution, and another with no real solutions.

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
D=b%5E2-4ac Start with the discriminant formula


D=k%5E2-4%282%29%289%29 Plug in a=2, b=k and c=9


D=k%5E2-8%289%29 Multiply 4 and 2 to get 8


D=k%5E2-72 Multiply 8 and 9 to get 72


So we'll use the equation D=k%5E2-72 to solve the following:

1) "find the values of k for which the quadratic 2x%5E2%2Bkx%2B9=0 has one real solution"


If a quadratic has one real solution, then the discriminant is equal to zero. So this means that D=0


D=k%5E2-72 Start with the previous equation


0=k%5E2-72 Plug in D=0


72=k%5E2 Add 72 to both sides


0%2B-sqrt%2872%29=k Take the square root of both sides


k=sqrt%2872%29 or k=-sqrt%2872%29 Break up the "plus/minus"


k=2%2Asqrt%2818%29 or k=-2%2Asqrt%2818%29 Simplify the square root


So if k=2%2Asqrt%2818%29 or k=-2%2Asqrt%2818%29, then the discriminant is equal to zero. This means that the equation 2x%5E2%2B2%2Asqrt%2818%29x%2B9=0 or 2x%5E2-2%2Asqrt%2818%29x%2B9=0 only has one real solution.




2) "find the values of k for which the quadratic 2x%5E2%2Bkx%2B9=0 no real solutions"


If a quadratic has no real solutions, this means that the discriminant is less than zero. In other words, D%3C0


D=k%5E2-72 Go back to the previous equation


k%5E2-72%3C0 Since D%3C0, this means that the right side is less than zero

k%5E2%3C72 Add 72 to both sides


k%3C0%2B-sqrt%2872%29 Take the square root of both sides


k%3Csqrt%2872%29 or k%3E-sqrt%2872%29 Break up the "plus/minus"


k%3C2%2Asqrt%2818%29 or k%3E-2%2Asqrt%2818%29 Simplify the square root


-2%2Asqrt%2818%29%3Ck%3C2%2Asqrt%2818%29 Recombine the two inequalities to form one compound inequality

So if "k" is in between -2%2Asqrt%2818%29 and 2%2Asqrt%2818%29, then the discriminant is less than zero. This means that if "k" is in between -2%2Asqrt%2818%29 and 2%2Asqrt%2818%29, then the quadratic 2x%5E2%2Bkx%2B9=0 will have no solutions