How to Solve Quadratic Equations
Quadratic equations appear throughout mathematics, physics, engineering, and business. Whether you are calculating the arc of a thrown ball, optimizing a product price for maximum revenue, or finding the dimensions of a rectangular area, the solution almost always involves solving an equation with an x² term. The good news is that there are three reliable methods for solving any quadratic equation, and each has situations where it shines.
This guide covers all three methods in detail — the quadratic formula, factoring, and completing the square — with step-by-step worked examples for each. You will also learn how to use the discriminant to predict the number and type of solutions before solving.
What Is a Quadratic Equation?
A quadratic equation is a second-degree polynomial equation in a single variable. Its standard form is:
ax² + bx + c = 0
a = coefficient of x² (must not be zero)
b = coefficient of x
c = constant term
The requirement that a cannot equal zero is what makes the equation quadratic. If a were zero, the x² term would vanish and you would have a linear equation (bx + c = 0), which has only one solution.
Every quadratic equation has exactly two solutions (also called roots or zeros), though they may be real or complex numbers, and the two solutions may be identical (a repeated root). The graph of a quadratic function y = ax² + bx + c is a parabola — a U-shaped curve that opens upward when a is positive and downward when a is negative.
Examples of Quadratic Equations
| Equation | a | b | c |
|---|---|---|---|
| 2x² + 5x - 3 = 0 | 2 | 5 | -3 |
| x² - 9 = 0 | 1 | 0 | -9 |
| -4x² + 8x = 0 | -4 | 8 | 0 |
| x² + 6x + 9 = 0 | 1 | 6 | 9 |
Method 1: The Quadratic Formula
The quadratic formula is the universal method that works on every quadratic equation. It directly computes both solutions from the coefficients a, b, and c:
x = (-b ± √(b² - 4ac)) / 2a
The ± symbol means you calculate two values: one using addition and one using subtraction. These give you the two solutions.
Example: Solve 2x² + 5x - 3 = 0
Identify coefficients: a = 2, b = 5, c = -3
Calculate the discriminant: b² - 4ac = 25 - 4(2)(-3) = 25 + 24 = 49
Apply the formula: x = (-5 ± √49) / (2 × 2) = (-5 ± 7) / 4
Solution 1: x = (-5 + 7) / 4 = 2/4 = 0.5
Solution 2: x = (-5 - 7) / 4 = -12/4 = -3
Verify: 2(0.5)² + 5(0.5) - 3 = 0.5 + 2.5 - 3 = 0 ✓
Try Our Quadratic Formula Calculator
Enter your coefficients a, b, and c to instantly solve any quadratic equation using the quadratic formula.
Use CalculatorExample: Solve 3x² - 2x + 5 = 0
Identify coefficients: a = 3, b = -2, c = 5
Calculate the discriminant: b² - 4ac = 4 - 60 = -56
Result: Since the discriminant is negative, there are no real solutions.
The complex solutions are x = (2 ± √(-56)) / 6 = (1 ± i√14) / 3
Method 2: Factoring
Factoring rewrites the quadratic expression as a product of two linear factors. If ax² + bx + c = (px + q)(rx + s) = 0, then either px + q = 0 or rx + s = 0, giving you the two solutions directly. This method is the fastest when it works, but not all quadratics factor neatly with integers.
Example: Solve x² + 5x + 6 = 0 by factoring
Find two numbers that multiply to 6 and add to 5: The numbers are 2 and 3
Write the factors: (x + 2)(x + 3) = 0
Set each factor to zero:
x + 2 = 0 → x = -2
x + 3 = 0 → x = -3
Verify: (-2)² + 5(-2) + 6 = 4 - 10 + 6 = 0 ✓
For equations where a is not 1, factoring requires a few more steps. The AC method is the most systematic approach.
Example: Solve 6x² + 11x - 10 = 0 using the AC method
Step 1: Multiply a × c: 6 × (-10) = -60
Step 2: Find two numbers that multiply to -60 and add to 11: 15 and -4
Step 3: Rewrite the middle term: 6x² + 15x - 4x - 10 = 0
Step 4: Factor by grouping: 3x(2x + 5) - 2(2x + 5) = 0
Step 5: Factor out the common binomial: (3x - 2)(2x + 5) = 0
Solutions: x = 2/3 and x = -5/2
Method 3: Completing the Square
Completing the square transforms the quadratic into a perfect square trinomial, making it easy to solve by taking square roots. This method is also the foundation for deriving the quadratic formula itself and for converting quadratic functions to vertex form.
Example: Solve x² + 8x + 7 = 0 by completing the square
Step 1: Move the constant to the right: x² + 8x = -7
Step 2: Take half of b (which is 8), square it: (8/2)² = 16
Step 3: Add 16 to both sides: x² + 8x + 16 = -7 + 16 = 9
Step 4: Write the left side as a perfect square: (x + 4)² = 9
Step 5: Take the square root of both sides: x + 4 = ±3
Solution 1: x = -4 + 3 = -1
Solution 2: x = -4 - 3 = -7
Verify: (-1)² + 8(-1) + 7 = 1 - 8 + 7 = 0 ✓
Example: Solve 2x² - 12x + 10 = 0 by completing the square
Step 1: Divide everything by a = 2: x² - 6x + 5 = 0
Step 2: Move the constant: x² - 6x = -5
Step 3: Half of -6 is -3, squared is 9: x² - 6x + 9 = -5 + 9 = 4
Step 4: Perfect square form: (x - 3)² = 4
Step 5: Take square roots: x - 3 = ±2
Solutions: x = 5 and x = 1
Try Our Root Calculator
Find square roots, cube roots, and nth roots used when solving quadratic equations.
Use CalculatorThe Discriminant and Nature of Solutions
The expression under the square root in the quadratic formula, b² - 4ac, is called the discriminant (often denoted by the Greek letter delta, Δ). It tells you everything about the number and type of solutions without actually solving the equation.
| Discriminant (b² - 4ac) | Number of Solutions | Type | Graph Behavior |
|---|---|---|---|
| Positive (> 0) | 2 distinct | Real | Parabola crosses x-axis twice |
| Zero (= 0) | 1 repeated | Real | Parabola touches x-axis once |
| Negative (< 0) | 2 distinct | Complex | Parabola does not touch x-axis |
Example: Classify solutions without solving
Equation: x² - 6x + 9 = 0
Discriminant: (-6)² - 4(1)(9) = 36 - 36 = 0 → One repeated real root
Equation: 2x² + 3x - 5 = 0
Discriminant: 9 - 4(2)(-5) = 9 + 40 = 49 → Two distinct real roots
Equation: x² + x + 1 = 0
Discriminant: 1 - 4(1)(1) = 1 - 4 = -3 → Two complex roots (no real solutions)
Practical Applications of Quadratic Equations
Quadratic equations model a surprising variety of real-world scenarios. Here are three examples that show how different fields use these equations.
Application 1: Projectile Motion — Nadia launches a rocket
Nadia launches a model rocket with an initial velocity of 96 feet per second from the ground. The height h (in feet) at time t (in seconds) is modeled by:
h = -16t² + 96t
When does the rocket hit the ground?
Set h = 0: -16t² + 96t = 0
Factor: -16t(t - 6) = 0
Solutions: t = 0 (launch) and t = 6 seconds (landing)
Maximum height: At the vertex, t = -96 / (2 × -16) = 3 seconds
h(3) = -16(9) + 96(3) = -144 + 288 = 144 feet
Application 2: Business Optimization — Carlos prices his product
Carlos sells handmade phone cases. At $25 each, he sells 200 units per month. For every $1 price increase, he sells 5 fewer units. What price maximizes revenue?
Let x = number of $1 increases. Price = 25 + x, Quantity = 200 - 5x
Revenue = (25 + x)(200 - 5x) = -5x² + 75x + 5000
Maximum at vertex: x = -75 / (2 × -5) = 7.5
Optimal price: $25 + $7.50 = $32.50
Maximum revenue: -5(7.5)² + 75(7.5) + 5000 = $5,281.25 per month
Application 3: Geometry — Elena designs a garden border
Elena has a rectangular garden that is 20 feet by 15 feet. She wants to add a uniform border of flowers around it. If she has enough flowers for 234 square feet of border, how wide should the border be?
Let w = border width. Total area = (20 + 2w)(15 + 2w)
Border area = Total area - Garden area
(20 + 2w)(15 + 2w) - 300 = 234
300 + 40w + 30w + 4w² - 300 = 234
4w² + 70w - 234 = 0 → 2w² + 35w - 117 = 0
Using the quadratic formula: w = (-35 + √(1225 + 936)) / 4 = (-35 + √2161) / 4
w = (-35 + 46.49) / 4 ≈ 2.87 feet (discard the negative solution)
Choosing the Right Method
Each method has strengths depending on the equation. Here is a quick decision guide:
| Method | Best When | Drawback |
|---|---|---|
| Quadratic Formula | Always works; use as default | More arithmetic steps |
| Factoring | Coefficients are small integers | Not all equations factor over integers |
| Completing the Square | Finding vertex form; deriving formulas | Tedious with non-integer coefficients |
A practical strategy is to first check if the equation factors easily (spend no more than 30 seconds looking for integer factors). If not, go straight to the quadratic formula. Use completing the square when you need the vertex form of the parabola for graphing or optimization problems.
Tips for Solving Quadratic Equations
- Always put the equation in standard form first. Move all terms to one side so you have ax² + bx + c = 0 before identifying a, b, and c. Skipping this step is the most common source of errors.
- Check the discriminant before solving. A quick calculation of b² - 4ac tells you what to expect. If it is negative, you know there are no real solutions, which can save you time on exams.
- Verify your solutions by substitution. Plug each solution back into the original equation. If both sides equal zero, your answer is correct. This takes 30 seconds and catches arithmetic mistakes.
- Watch your signs carefully. The most common errors involve mishandling negative values for b or c. Write out each step explicitly rather than doing multiple operations in your head.
- Look for common factors first. Before applying any method, check if all terms share a common factor. Simplifying 6x² + 12x - 18 = 0 to x² + 2x - 3 = 0 by dividing by 6 makes every method easier.
Common Mistakes to Avoid
- Forgetting the ± in the quadratic formula. The formula gives two solutions. Writing only one (using + or -) means you miss half the answer. Always compute both branches.
- Dividing by 2a incorrectly. The entire numerator (-b ± √(b² - 4ac)) is divided by 2a, not just part of it. Use parentheses to keep the calculation organized: (-b ± √Δ) / (2a).
- Misidentifying b when it is negative. In the equation x² - 7x + 12 = 0, b = -7 (not 7). When you substitute into -b, you get -(-7) = 7. Double negatives require extra care.
- Forgetting to set the equation equal to zero before factoring. The zero product property only works when the product equals zero. You cannot factor x² + 5x = 6 directly; first rewrite it as x² + 5x - 6 = 0.
- Not simplifying before solving. An equation like 4x² + 8x + 4 = 0 simplifies to x² + 2x + 1 = 0 (divide by 4), which factors easily as (x + 1)² = 0. Skipping simplification makes the problem harder than it needs to be.
Try Our Scientific Calculator
Perform complex calculations including square roots and exponents for quadratic equation solving.
Use CalculatorFrequently Asked Questions
A quadratic equation is a polynomial equation where the highest power of the variable is 2. The standard form is ax² + bx + c = 0, where a cannot equal zero. If a were zero, the x² term would disappear and you would have a linear equation. Quadratic equations can have terms with x², x, and constants, but no x³ or higher powers.
Yes. When the discriminant (b² - 4ac) is negative, the equation has no real number solutions because you cannot take the square root of a negative number in real arithmetic. The two solutions in this case are complex numbers involving the imaginary unit i. Graphically, this means the parabola does not cross the x-axis at all — it floats entirely above or below it.
The quadratic formula works on every quadratic equation, so it is the safest choice when you are unsure whether factoring will work. Factoring is faster when the equation has integer roots, which happens when the discriminant is a perfect square. If you try factoring for more than a minute without finding factors, switch to the quadratic formula to save time.
Standard form is ax² + bx + c = 0 with all terms on one side and zero on the other. If your equation looks like 3x² + 5 = 2x, rearrange it to 3x² - 2x + 5 = 0 before identifying a, b, and c. Always ensure a is positive for consistency, though the formula works with negative a values as well. Every quadratic equation can be rearranged into standard form.
The roots (solutions) of a quadratic equation correspond to the x-intercepts of the parabola y = ax² + bx + c. If the equation has two distinct real roots, the parabola crosses the x-axis at two points. One repeated root means the parabola touches the x-axis at exactly one point (the vertex). No real roots means the parabola never touches the x-axis.
Yes. The quadratic formula often produces answers involving square roots that are irrational numbers. For example, solving x² - 2 = 0 gives x = plus or minus the square root of 2, which is approximately 1.414. Leaving the answer in radical form (with the square root sign) is the exact answer, while any decimal approximation is rounded and therefore slightly imprecise.
Quadratic equations model any situation where a quantity depends on the square of a variable. Projectile motion follows a parabolic path described by quadratic equations. Business revenue is a quadratic function of price when demand varies linearly. Area calculations involving unknown dimensions often produce quadratic equations. Engineers, physicists, and economists use quadratics routinely in their work.
Sources & References
- Math Is Fun — Interactive guide to quadratic equations with visual examples: mathsisfun.com
- Wolfram MathWorld — Comprehensive mathematical reference on quadratic equations: mathworld.wolfram.com
- Purplemath — Step-by-step quadratic formula lessons with worked examples: purplemath.com
CalculatorGlobe Team
Content & Research Team
The CalculatorGlobe team creates in-depth guides backed by authoritative sources to help you understand the math behind everyday decisions.
Related Calculators
Quadratic Formula Calculator
Solve any quadratic equation instantly using the quadratic formula.
Scientific Calculator
Perform complex mathematical calculations including polynomials.
Root Calculator
Find square roots and nth roots used in quadratic solutions.
Exponent Calculator
Calculate powers and exponents in polynomial expressions.
Half-Life Calculator
Apply quadratic and exponential equations to radioactive decay.
NPV Calculator
Use polynomial equations in net present value financial analysis.
Related Articles
How to Calculate Standard Deviation: Step-by-Step
Master standard deviation calculation with a clear step-by-step method, understand population vs sample formulas, and see practical real data examples.
Understanding Percentages: Practical Applications
Master percentage calculations for discounts, tips, grades, and growth rates with clear formulas, step-by-step examples, and common conversion shortcuts.
The Golden Ratio in Nature, Art, and Design
Discover the golden ratio and its appearances in nature, architecture, art, and modern design with mathematical proofs and visual examples explained clearly.
How to Calculate Area and Volume of Common Shapes
Calculate area and volume for circles, rectangles, triangles, spheres, cylinders, and cones with formulas, diagrams, and practical measurement applications.
Disclaimer: This calculator is for informational and educational purposes only. Results are estimates and may not reflect exact values.
Last updated: February 23, 2026