Practice · Grade 9 Algebra
Quadratic Equations — Practice Problems
Train the quadratic formula with problems of rising difficulty plus a boss question. Each comes with discriminant, hint and full solution. Free, grade 9.
Q1 of 6
0 correct
Solve the quadratic equation.
x² − 5x + 6 = 0
Quick answer
What's the best way to practice quadratic equations?
Work through at least five problems of increasing difficulty. Start from the standard form ax² + bx + c = 0 and the quadratic formula x = (−b ± √(b² − 4ac)) / (2a). Always compute the discriminant D = b² − 4ac first, because it tells you whether you get two, one or no solutions. Then check your answer with Vieta's formulas (x₁ + x₂ = −b/a, x₁ · x₂ = c/a), and reveal a hint once more rather than once too few — understanding beats guessing.
HowTo
A 4-step solving strategy
This order works for every quadratic equation — whether a = 1 or a ≠ 1.
- 1Step 1 of 4
Put it in standard form and read off a, b, c
Move everything to one side so it reads ax² + bx + c = 0. Then read a, b, c with their signs. For x² − 5x + 6 = 0 that's a = 1, b = −5, c = 6.
- 2Step 2 of 4
Compute the discriminant
Plug into D = b² − 4ac. The sign decides: D > 0 gives two solutions, D = 0 one double root, D < 0 no real solution. Use brackets: (−5)² = 25.
- 3Step 3 of 4
Substitute into the quadratic formula
x = (−b ± √D) / (2a). Mind the denominator 2a — if a = 2 you divide by 4, not by 2. The ± produces the two solutions.
- 4Step 4 of 4
Work out both solutions and check
Compute the + and the − case separately. Verify with Vieta: x₁ + x₂ must equal −b/a and x₁ · x₂ must equal c/a. Only then is the problem safely solved.
Examples
Worked examples with a full solution path
Four typical problem types from grade 9 tests. Try each one yourself first, then compare with the solution.
Easy
Solve: x² − 2x − 15 = 0
a = 1, b = −2, c = −15
D = (−2)² − 4 · 1 · (−15) = 4 + 60 = 64
x = (2 ± √64)/2 = (2 ± 8)/2
x₁ = 5, x₂ = −3
Check (Vieta): 5 + (−3) = 2 = −b/a; 5 · (−3) = −15 = c/a ✓
Standard form with a = 1. The negative c makes D large — both solutions are real and distinct.
Medium
Solve: x² − 10x + 25 = 0
a = 1, b = −10, c = 25
D = (−10)² − 4 · 1 · 25 = 100 − 100 = 0
x = −b/(2a) = 10/2
x = 5 (double root)
Check: 5² − 10 · 5 + 25 = 25 − 50 + 25 = 0 ✓
D = 0 means a double root — the parabola just touches the x-axis. It's the square (x − 5)².
Medium
Solve: 3x² − 12x = 0
a = 3, b = −12, c = 0
D = (−12)² − 4 · 3 · 0 = 144
x = (12 ± 12)/6
x₁ = 4, x₂ = 0
Check: 3 · 4² − 12 · 4 = 48 − 48 = 0; 3 · 0² − 12 · 0 = 0 ✓
When c is missing, one solution is always 0. Faster: 3x(x − 4) = 0 → x = 0 or x = 4.
Hard
Boss: 2x² + 5x − 3 = 0
a = 2, b = 5, c = −3
D = 5² − 4 · 2 · (−3) = 25 + 24 = 49
x = (−5 ± √49)/4 = (−5 ± 7)/4
x₁ = 0.5, x₂ = −3
Check (Vieta): 0.5 + (−3) = −2.5 = −b/a; 0.5 · (−3) = −1.5 = c/a ✓
a ≠ 1 and a negative c — clean sign handling pays off here. The denominator is 2a = 4.
Pitfalls
Common mistakes — and how to avoid them
These five traps show up in almost every test on quadratic equations.
Dropping the sign of b in the formula
The formula has −b. With b = −5 that becomes +5. And (−5)² = +25, not −25 — brackets save you.
Not converting to standard form first
Move everything to one side (ax² + bx + c = 0) before reading off a, b, c. Otherwise even the coefficients are wrong.
Dividing by 2 instead of 2a
The denominator is 2a. If a = 2 you divide by 4. Forget the a and both solutions are wrong.
Misreading the discriminant
D < 0 means no real solution — not "x = 0". D = 0 gives exactly one (double) root, not two.
Giving only one of the two solutions
When D > 0 the ± always yields two values. Compute the + and the − case separately and write both down.
Study
Practice with a plan — three short tips
Discriminant first, always
Compute D before you substitute into the formula. That tells you instantly whether you get two, one or no solutions — and catches errors early.
Solve first, then look at the answer
Write down your full solution before revealing the hint. Active recall is three to four times more effective for learning than passive reading.
Check every solution with Vieta
x₁ + x₂ = −b/a and x₁ · x₂ = c/a take ten seconds to verify and catch almost any sign or arithmetic slip.
FAQ
Frequently asked questions about practising
Glossary
Terms in one sentence
- Quadratic equation
- An equation of the form ax² + bx + c = 0 with a ≠ 0; the variable appears to the second power.
- Quadratic formula
- The solution formula x = (−b ± √(b² − 4ac)) / (2a), also called the a-b-c formula.
- Discriminant
- The term D = b² − 4ac under the root, which determines the number of solutions.
- Double root
- A solution that counts twice; it occurs exactly when D = 0.
- Standard form
- The form ax² + bx + c = 0 with everything on one side of the equation.
- Vieta's formulas
- x₁ + x₂ = −b/a and x₁ · x₂ = c/a — handy for checking your answer.
- Boss question
- The last and hardest problem of a practice set, combining several difficulties.