Find the constant.Find the constant k so that the quadratic equation 2x2 + 5x - k = 0 has two real solutions.

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

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A quadratic equation ax^2 + bx + c = 0 has two real roots if b^2 > 4ac

For 2x^2 + 5x - k = 0 to have two real roots:

5^2 > 4*2*(-k)

=> 25 > -8k

=> k > -25/8

The term k can take any value greater than -25/8

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

When a quadratic has 2 real solutions, it's discriminant delta is positive or equal to zero.

delta > = 0

delta = b^2 - 4ac, where a,b,c are the coefficients of the quadratic.

Since it is not specified if the roots are different or they are equal, we'll consider both cases:

delta > 0

b^2 - 4ac > 0

We'll identify a,b,c:

2x^2 + 5x - k = 0

a = 2 , b = 5 , c = -k

delta = 25 + 8k

25 + 8k > 0

8k > -25

k > -25/8

Therefore, if the values of k are in the interval (-25/8 ; +infinite), the quadratic has 2 different real roots.

For delta = 0 => x1 = x2

25 + 8k = 0

8k = -25

k = -25/8

If k = -25/8, the roots of the quadratic are equal and real.

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