1.3 The roots of a quadratic equation are \( x=\frac{-10 \pm \sqrt{100-4 k^{2}}}{2 k} \). Calculate the values of \( k \) for which the roots are: 1.3.1 equal 1.3 .2 non-real

We are given the roots of a quadratic equation in the form   x = (–10 ± √(100 – 4k²)) / (2k). This expression is based on the quadratic formula for an equation of the type   kx² + 10x + k = 0      (with k ≠ 0). To decide when the roots are equal or non-real, we need to examine the discriminant, which in this formula is the expression under the square root. Step 1. Find the Discriminant The discriminant, Δ, is given by   Δ = 100 – 4k². Step 2. Conditions for the Types of Roots 1. Equal Roots: The roots are equal when Δ = 0.   100 – 4k² = 0   => 4k² = 100   => k² = 25   => k = 5 or k = –5. So, the roots are equal when k = 5 or k = –5. 2. Non-real (Complex) Roots: The roots are non-real when Δ < 0.   100 – 4k² < 0   => 4k² > 100   => k² > 25   => |k| > 5   => k < –5 or k > 5. Thus, the roots are non-real when |k| > 5 (that is, when k < –5 or k > 5). Note: We exclude k = 0 because the quadratic formula given would be undefined (division by zero), and the original quadratic equation would no longer be quadratic. Final Answers: 1.3.1 Equal roots occur when k = 5 or k = –5. 1.3.2 Non-real roots occur when |k| > 5 (i.e., k < –5 or k > 5).