x%5E2%2B(k%2B1)x%2B1%3D0-has-equal-roots.jpeg "(k+4)x^2+(k+1)x+1=0 Has Equal Roots")
Determining the Value of 'k' for a Quadratic Equation with Equal Roots
A quadratic equation of the form ax² + bx + c = 0 has equal roots when its discriminant, denoted by Δ, is equal to zero. The discriminant is calculated as:
Δ = b² - 4ac
In our given equation, (k+4)x² + (k+1)x + 1 = 0, we have:
- a = (k+4)
- b = (k+1)
- c = 1
To find the value of k for which the equation has equal roots, we set the discriminant to zero:
(k+1)² - 4(k+4)(1) = 0
Expanding and simplifying the equation:
- k² + 2k + 1 - 4k - 16 = 0
- k² - 2k - 15 = 0
Factoring the quadratic equation:
(k - 5)(k + 3) = 0
Therefore, the possible values of k are:
- k = 5
- k = -3
Conclusion:
We have determined that the quadratic equation (k+4)x² + (k+1)x + 1 = 0 will have equal roots when k = 5 or k = -3. This means that for these specific values of k, the equation will have only one distinct solution.