Determining the Values of k for Which kx2 - 2x - 7 has No Real Roots

When dealing with quadratic equations of the form ax2 bx c 0, the discriminant is a crucial tool for determining the nature of the roots. The discriminant of a quadratic equation is given by the formula D b2 - 4ac. This value tells us the number of real roots the equation has:

If D 0, the equation has two distinct real roots. If D 0, the equation has exactly one real root (a repeated root). If D 0, the equation has no real roots (the roots are complex).

Let's explore how to apply this concept to the specific quadratic equation kx2 - 2x - 7, focusing on determining the values of k for which the equation has no real roots.

Understanding the Discriminant

Given the quadratic equation kx2 - 2x - 7, we can identify the coefficients as follows:

a k b -2 c -7

Using these coefficients, we can now compute the discriminant:

D b2 - 4ac

Substituting the values from our specific quadratic equation:

D (-2)2 - 4(k)(-7)

D 4 28k

Conditions for No Real Roots

To have no real roots, the discriminant must be less than zero:

4 28k 0

Let's solve this inequality for k:

Subtract 4 from both sides:

28k -4

Divide both sides by 28:

k -4/28

Reduce the fraction:

k -1/7

Solving this inequality shows that the quadratic equation kx2 - 2x - 7 has no real roots when k is less than -1/7, and it has real roots when k is greater than or equal to -1/7.

Exploring Additional Solution Methods

Let's explore another way to solve for the values of k:

Another approach involves completing the square for the quadratic expression kx2 - 2x - 7:

kx2 - 2x - 7 0

x2 - (2/k)x 7/k

Add (2/k)2 to both sides to complete the square:

x2 - (2/k)x (1/k2) 7/k (1/k2)

(x - 1/k)2 (7 1)/k2

(x - 1/k)2 (8 1)/k2

(x - 1/k)2 (8 1)/k2

(x - 1/k)2 (7 1)/k2

For the equation to have no real roots, the right side must be less than zero:

(7 1)/k2 0

This is not possible since (7 1) is positive and dividing by k2 will always result in a positive value for any real k.

Conclusion

In conclusion, for the quadratic equation kx2 - 2x - 7 to have no real roots, the value of k must be less than -1/7. This is because the discriminant of the equation needs to be less than zero for the quadratic to have no real roots.

Understanding the discriminant and its role in determining the nature of the roots in a quadratic equation is a valuable skill in algebra and provides a deeper insight into the behavior of quadratic expressions. Whether you use the discriminant formula or complete the square, the key is to manipulate the equation appropriately and interpret the resulting values to determine the required conditions.

Keywords: Quadratic equation, discriminant, real roots, no real roots