10-5 Solving Quadratic Equations by Completing the Square — Practice Questions

1. 1. After completing the square, an equation becomes $(x + 5)^2 = -4$. How many real solutions does this equation have?

   • A. Two real solutions
   • B. Exactly one real solution
   • C. No real solutions
   • D. Infinitely many solutions

2. 2. Solve $x^2 + 6x + 11 = 0$ by completing the square. After factoring, the equation becomes $(x + 3)^2 = $ ___. Since this value is negative, the number of real solutions is ___.

3. 3. Which equation, after completing the square, results in NO real solutions?

   • A. $x^2 + 2x - 5 = 0$
   • B. $x^2 - 6x + 5 = 0$
   • C. $x^2 + 2x + 6 = 0$
   • D. $x^2 - 4x + 3 = 0$

4. 4. Solve $3x^2 - 6x + 9 = 0$ by completing the square. After dividing and isolating, the equation becomes $(x - 1)^2 = $ ___, which means there are ___ real solutions.

5. 5. After completing the square on $x^2 - 8x + 20 = 0$, what is the value of $d$ in $(x - 4)^2 = d$, and what does it tell us?

   • A. $d = 4$; two real solutions
   • B. $d = -4$; no real solutions
   • C. $d = 0$; exactly one real solution
   • D. $d = 36$; two real solutions

6. 6. A ball's height is modeled by $h(t) = -16t^2 + 64t + 10$. After completing the square, the vertex form is $h(t) = -16(t-2)^2 + 74$. The maximum height reached by the ball is ___ feet.

7. 7. A projectile's height is given by $h(t) = -16(t - 3)^2 + 45$. At what time $t$ does the projectile reach its maximum height?

   • A. $t = 45$ seconds
   • B. $t = 3$ seconds
   • C. $t = -3$ seconds
   • D. $t = 16$ seconds

8. 8. A water arc follows $h(x) = -0.25x^2 + 2x$. After completing the square, which vertex form is correct?

   • A. $h(x) = -0.25(x - 4)^2 + 4$
   • B. $h(x) = -0.25(x - 2)^2 + 4$
   • C. $h(x) = -0.25(x - 4)^2 + 2$
   • D. $h(x) = -0.25(x - 2)^2 + 2$

9. 9. A ball's height is modeled by $h(t) = -16t^2 + 48t + 7$. Factor out $-16$ from the first two terms and complete the square. The maximum height of the ball is ___ feet.

10. 10. Using the vertex form $h(t) = -16(t - 2)^2 + 36$, which equation correctly sets up finding when the object hits the ground?

    • A. $-16(t-2)^2 = 36$
    • B. $-16(t-2)^2 + 36 = 0$
    • C. $16(t-2)^2 + 36 = 0$
    • D. $(t-2)^2 = 36$