Question

Determine the value of k with the specified condition.

x²+4x+k=0 - the roots are equal
kx²−5x−2=0 - the roots are equal
kx² +4x+6=0 - the roots are imaginary
x²+5x−k=0 - real double roots
kx²+3kx+2k+1=0 - two equal real roots

Answer 1

To determine the value of k for various quadratic equations with specified root conditions, the discriminant (b² - 4ac) is utilized. Equal or real double roots require the discriminant to be zero, while imaginary roots require it to be negative.

Step-by-step explanation:

The question involves determining the value of k for a series of quadratic equations under various conditions. To find these values, we use the discriminant from the quadratic formula, which is b2 - 4ac, where ax2 + bx + c = 0 is the standard form of a quadratic equation.

For equal roots, the discriminant must be 0; for imaginary roots, it must be less than 0; and for real double roots, it must also be 0. We can calculate the value of k by setting the discriminant equal to the desired condition and solving for k.

• x2+4x+k=0 - the roots are equal (k=4).
• kx2−5x−2=0 - the roots are equal (k=5/4).
• kx2+4x+6=0 - the roots are imaginary (k<0).
• x2+5x−k=0 - real double roots (k= 6.25).
• kx2+3kx+2k+1=0 - two equal real roots (k=-1).