Always positive or always negative questions with answers

Questions

1. Question 1x² + bx + c

   Determine whether $x^2 + 8x + 15$ is always positive, always negative or neither for all real values of $x$.

   Show the worked solution

   1. Step 1: Complete the square$x^2 + 8x + 15 = (x + 4)^2 - 16 + 15$$= (x + 4)^2 - 1$
   2. Step 2: A square is never negative$(x + 4)^2 \geq 0$$x^2 + 8x + 15 \geq -1$So it is neither: it is positive for some x and negative for others.
2. Question 2x² + bx + c

   Determine whether $x^2 + 2x - 2$ is always positive, always negative or neither for all real values of $x$.

   Show the worked solution

   1. Step 1: Complete the square$x^2 + 2x - 2 = (x + 1)^2 - 1 - 2$$= (x + 1)^2 - 3$
   2. Step 2: A square is never negative$(x + 1)^2 \geq 0$$x^2 + 2x - 2 \geq -3$So it is neither: it is positive for some x and negative for others.
3. Question 3c + bx − x²

   Determine whether $2 - 2x - x^2$ is always positive, always negative or neither for all real values of $x$.

   Show the worked solution

   1. Step 1: Take out −1, then complete the square$2 - 2x - x^2 = -(x^2 + 2x) + 2$$= -[(x + 1)^2 - 1] + 2$$= -(x + 1)^2 + 1 + 2$$= -(x + 1)^2 + 3$
   2. Step 2: A square is never negative$(x + 1)^2 \geq 0$$2 - 2x - x^2 \leq 3$So it is neither: it is positive for some x and negative for others.
4. Question 4c + bx − x²

   Determine whether $-2 - 2x - x^2$ is always positive, always negative or neither for all real values of $x$.

   Show the worked solution

   1. Step 1: Take out −1, then complete the square$-2 - 2x - x^2 = -(x^2 + 2x) - 2$$= -[(x + 1)^2 - 1] - 2$$= -(x + 1)^2 + 1 - 2$$= -(x + 1)^2 - 1$
   2. Step 2: A square is never negative$(x + 1)^2 \geq 0$$-2 - 2x - x^2 \leq -1$So it is always negative.
5. Question 5Read the completed square

   Determine whether $3(x + 1)^2 - 4$ is always positive, always negative or neither for all real values of $x$.

   Show the worked solution

   1. Step 1: A square is never negative$(x + 1)^2 \geq 0$$3(x + 1)^2 \geq 0$
   2. Step 2: So compare the whole expression with the number added$3(x + 1)^2 - 4 \geq -4$So it is neither: it is positive for some x and negative for others.
6. Question 6Read the completed square

   Determine whether $-2(x - 3)^2 - 1$ is always positive, always negative or neither for all real values of $x$.

   Show the worked solution

   1. Step 1: A square is never negative$(x - 3)^2 \geq 0$$-2(x - 3)^2 \leq 0$
   2. Step 2: So compare the whole expression with the number added$-2(x - 3)^2 - 1 \leq -1$So it is always negative.