Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
These problems focus on quadratic equations whose solutions are complex numbers. They address Common Core standards N-CN.C.7 and A-REI.B.4b and rely on the discriminant, the quadratic formula, and completing the square.
Early problems classify equations and solve simple cases; later ones use fractions, parameters, graphs, and real-world models in which a quadratic never reaches a target value.
Standards: N-CN.C.7, A-REI.B.4b · Level: Grade 11 Algebra 2
Problem 1 Discriminant and type of solutions ★★★
Find the discriminant of each equation and state the number and type of solutions.
- x2 − 3x − 10 = 0
- x2 − 8x + 16 = 0
- 2x2 + 3x + 4 = 0
- 3x2 − 2x − 7 = 0
Problem 2 Solving with square roots ★★★
Solve each equation. Write the solutions in terms of i.
- x2 = −49
- x2 + 36 = 0
- 3x2 + 75 = 0
Problem 3 Solutions from a squared binomial ★★★
Solve each equation.
- (x + 4)2 = −25
- (x − 3)2 + 49 = 0
Problem 4 Quadratic formula with a negative discriminant ★★★
Use the quadratic formula to solve x2 + 10x + 41 = 0.
Problem 5 Reading the graph ★★★
The graph of y = x2 − 8x + 19 is shown. State how many real solutions x2 − 8x + 19 = 0 has, give the sign of the discriminant, and find all solutions.
Problem 6 Completing the square ★★★
Solve x2 − 12x + 45 = 0 by completing the square.
Problem 7 Fractions in the solution ★★★
Solve 3x2 + 5x + 4 = 0.
Problem 8 A parabola that opens downward ★★★
The graph of y = −x2 + 2x − 6 is shown. Explain why the equation −x2 + 2x − 6 = 0 has no real solutions, then solve it.
Problem 9 A solution with a reduced fraction ★★★
Solve 5x2 − 4x + 1 = 0.
Problem 10 Reducing a radical solution ★★★
Solve 3x2 − 6x + 10 = 0 and write the answer in simplest form.
Problem 11 Choosing a constant ★★★
For which values of k does x2 − 8x + k = 0 have non-real solutions? For which value does it have exactly one real solution?
Problem 12 Plotting complex solutions ★★★
Find both solutions of x2 + 6x + 18 = 0, mark them as points on the complex-plane grid, and confirm the sum and product relationships between the solutions and the coefficients.
Problem 13 Building an equation from solutions ★★★
Write a quadratic equation with real coefficients and leading coefficient 1 whose solutions are 7 + 2i and 7 − 2i.
Problem 14 Which equations have non-real solutions ★★★
Without solving, decide which equations have non-real solutions.
- x2 − 5x + 7 = 0
- 2x2 − x − 6 = 0
- 9x2 − 12x + 4 = 0
- 4x2 + x + 3 = 0
Problem 15 Completing the square with a leading coefficient ★★★
Solve each equation by completing the square.
- x2 + 6x + 25 = 0
- 2x2 − 12x + 26 = 0
Problem 16 Will the ball reach the height? ★★★
A soccer ball kicked from a 5-foot platform has height h(t) = −16t2 + 40t + 5 feet after t seconds. A banner hangs 40 feet above the ground. Show with the discriminant that the ball never reaches the banner, and find the complex solutions of h(t) = 40.
Problem 17 Profit that cannot happen ★★★
A school fundraiser models profit as P(n) = −3n2 + 36n − 90 dollars when n hundred flyers are printed. Show that the profit can never reach $100, and find the greatest possible profit.
Problem 18 Sum and product of solutions ★★★
Solve 2x2 − 10x + 17 = 0. Then find the sum and product of the solutions and compare with −b/a and c/a.
Problem 19 Ranges of a parameter ★★★
For x2 + kx + 16 = 0, find all values of k that give (a) non-real solutions, (b) one real solution, (c) two real solutions.
Problem 20 Finding a missing coefficient ★★★
One solution of x2 + bx + c = 0, with b and c real, is 6 − 4i. Find the other solution, b, and c.
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