Nature of roots questions with answers

Questions

1. Question 1Discriminant and nature of roots

   Find the discriminant of $4x^2 + 4x + 1 = 0$ and state the nature of its roots.

   Show the worked solution

   1. Step 1: The discriminant: b² − 4ac$b^2 - 4ac = 4^2 - 4(4)(1)$$b^2 - 4ac = 0$
   2. Step 2: Compare the discriminant with 0It is 0, so there are two real, equal roots.
2. Question 2Equal roots: find k

   The equation $4x^2 + kx + 1 = 0$ has two equal roots. Find the possible values of $k$.

   Show the worked solution

   1. Step 1: Which condition on the discriminant gives two equal roots?$b^2 - 4ac = 0$
   2. Step 2: Substitute a, b and c, then make k² the subject$k^2 - 4(4)(1) = 0$$k^2 - 16 = 0$$k^2 = 16$
   3. Step 3: Square-root both sides: two answers$k = 4$ or $k = -4$
3. Question 3Range of k

   Find the range of values of $k$ for which $x^2 - 3x + k = 0$ has two real, distinct roots.

   Show the worked solution

   1. Step 1: Which condition on the discriminant gives two real, distinct roots?$b^2 - 4ac > 0$
   2. Step 2: Substitute a, b and c, then expand$(-3)^2 - 4(1)(k) > 0$$9 - 4k > 0$
   3. Step 3: Solve: dividing by a negative number reverses the sign$-4k > -9$$k < \dfrac{9}{4}$
4. Question 4Equal roots: k in two coefficients

   The equation $(k + 1)x^2 + 2k + 8 = 0$, where $k$ is a constant, has two equal roots. Find the value of $k$.

   Show the worked solution

   1. Step 1: Which condition on the discriminant gives two equal roots?$b^2 - 4ac = 0$
   2. Step 2: Substitute a, b and c (each may contain k)$(0)^2 - 4(k + 1)(2k + 8) = 0$
   3. Step 3: Expand and collect: a quadratic in k$-8k^{2} - 40k - 32 = 0$
   4. Step 4: Divide by the common factor (a negative number, so the x² term becomes positive)$k^{2} + 5k + 4 = 0$
   5. Step 5: Factorise$(k + 4)(k + 1) = 0$
   6. Step 6: Solve$k = -4$ or $k = -1$
   7. Step 7: A value that makes the coefficient of x² zero gives no quadratic: reject itWhen $k = -1$ the coefficient of $x^2$ is $0$, so reject $k = -1$.$k = -4$
5. Question 5Range of k: a quadratic inequality in k

   Find the range of values of $k$ for which the equation $x^2 + (k + 1)x + k + 9 = 0$ has two real, distinct roots.

   Show the worked solution

   1. Step 1: Which condition on the discriminant gives two real, distinct roots?$b^2 - 4ac > 0$
   2. Step 2: Substitute a, b and c (each may contain k)$(k + 1)^2 - 4(1)(k + 9) > 0$
   3. Step 3: Expand and collect: a quadratic in k$k^{2} - 2k - 35 > 0$
   4. Step 4: Factorise$(k + 5)(k - 7) > 0$
   5. Step 5: Find where it is 0$k = -5$ or $k = 7$
   6. Step 6: Sketch the ∪-shaped curve in k through the roots: which part do you want?The curve is above the axis outside the roots.
   7. Step 7: Write the range of k$k<-5 \text{ or } k>7$