Quadratic Formula Calculator — Free Online Equation Solver

How to Use the Quadratic Formula Calculator

1. Enter coefficient a (x² term): This is the number in front of x². It determines whether the parabola opens upward (positive) or downward (negative) and how wide or narrow it is. The default value is 1. This coefficient must not be zero — if a = 0, the equation becomes linear, not quadratic.
2. Enter coefficient b (x term): This is the number in front of x. It affects the horizontal position of the vertex and the axis of symmetry. The default value is -5. It can be positive, negative, or zero.
3. Enter coefficient c (constant): This is the constant term that has no x attached. It is the y-intercept of the parabola (the value of y when x = 0). The default is 6. It can be any real number.
4. Review the solutions: The right panel instantly shows the number and nature of roots (two real, one repeated, or two complex), the actual root values, the discriminant, the vertex coordinates, the vertex form of the equation, the axis of symmetry, and whether the parabola opens up or down. All results update in real time as you adjust coefficients.

The default equation x² - 5x + 6 = 0 factors to (x - 2)(x - 3) = 0, giving roots x = 2 and x = 3. Experiment with different coefficients to see how the discriminant, vertex, and root nature change. Try b² - 4ac = 0 for a single repeated root, or make the discriminant negative to see complex conjugate roots.

The Quadratic Formula

Variables Explained

• a (Leading Coefficient): Controls the width and direction of the parabola. Larger |a| means a narrower parabola. The sign of a determines whether the parabola opens upward (a > 0, minimum vertex) or downward (a < 0, maximum vertex).
• b (Linear Coefficient): Together with a, determines the axis of symmetry at x = -b/(2a). Changing b shifts the vertex horizontally and affects which direction the parabola is shifted from the y-axis.
• c (Constant Term): The y-intercept of the parabola. When x = 0, y = c. Increasing c shifts the parabola upward; decreasing c shifts it downward.
• D (Discriminant): The value b² - 4ac determines root nature. D > 0 means two distinct real roots, D = 0 means one repeated root, D < 0 means complex conjugate roots.

Step-by-Step Example

Solve 2x² + 3x - 5 = 0:

1. Identify coefficients: a = 2, b = 3, c = -5
2. Calculate the discriminant: D = 3² - 4(2)(-5) = 9 + 40 = 49
3. Since D > 0, there are two distinct real roots
4. x = (-3 ± sqrt(49)) / (2 x 2) = (-3 ± 7) / 4
5. x₁ = (-3 + 7) / 4 = 4/4 = 1
6. x₂ = (-3 - 7) / 4 = -10/4 = -2.5

Verify: 2(1)² + 3(1) - 5 = 2 + 3 - 5 = 0. Also: 2(-2.5)² + 3(-2.5) - 5 = 12.5 - 7.5 - 5 = 0. Both roots check out.

Practical Examples

Example 1: James Models Projectile Motion

James launches a ball upward from a 6-foot height with an initial velocity of 40 ft/s. The height equation is h(t) = -16t² + 40t + 6. He wants to find when the ball hits the ground (h = 0), so he solves -16t² + 40t + 6 = 0:

• a = -16, b = 40, c = 6
• D = 40² - 4(-16)(6) = 1600 + 384 = 1984
• t = (-40 ± sqrt(1984)) / (2 x -16) = (-40 ± 44.54) / -32
• t₁ = (-40 + 44.54) / -32 = -0.14 s (not physical — before launch)
• t₂ = (-40 - 44.54) / -32 = 2.64 s

The ball hits the ground after approximately 2.64 seconds. The negative root is discarded because negative time has no physical meaning in this context. The vertex occurs at t = -40/(2 x -16) = 1.25 seconds, when the ball reaches its maximum height of h(1.25) = 31 feet.

Example 2: Nina Optimizes Product Pricing

Nina's company sells widgets. Through market research, the demand function is q = 100 - 2p (quantity sold at price p). Revenue is R = p x q = p(100 - 2p) = -2p² + 100p. She wants to find the price that maximizes revenue:

• Revenue equation: R = -2p² + 100p (a = -2, b = 100)
• Vertex (maximum, since a < 0): p = -b/(2a) = -100/(2 x -2) = 25
• Maximum revenue: R(25) = -2(625) + 100(25) = -1250 + 2500 = $1,250

Nina should price each widget at $25 to maximize revenue at $1,250 per period. At this price, she sells 100 - 2(25) = 50 units. This is a classic application of quadratic optimization in business.

Example 3: Carlos Designs a Garden

Carlos has 60 meters of fencing to enclose a rectangular garden along a wall (needing only 3 sides). If one side has length x, the other two sides are each (60 - x)/2. The area is A = x(60 - x)/2 = -x²/2 + 30x. He wants to maximize the area:

• Area equation: A = -0.5x² + 30x (a = -0.5, b = 30)
• Maximum at x = -30/(2 x -0.5) = 30 meters
• Other sides: (60 - 30)/2 = 15 meters each
• Maximum area: -0.5(900) + 30(30) = -450 + 900 = 450 m²

Carlos should make the garden 30 meters along the wall and 15 meters deep, yielding a maximum area of 450 square meters. The vertex of the parabola gives the optimal dimensions.

Discriminant Analysis Reference Table

Tips and Complete Guide

When to Use Each Solving Method

For quadratic equations with small integer coefficients that factor cleanly (like x² - 5x + 6 = 0), factoring is the fastest method. For equations where you need the vertex (optimization problems), completing the square or using the vertex formula h = -b/(2a) is most direct. For all other cases, especially when the discriminant is not a perfect square or when coefficients are decimals, the quadratic formula is the most reliable approach. Our calculator uses the formula internally and displays all relevant information regardless of the equation type.

Understanding the Parabola Graph

Every quadratic equation represents a parabola when graphed. The vertex is the turning point, and the axis of symmetry divides the parabola into two mirror-image halves. The y-intercept is at (0, c), and the x-intercepts (if real roots exist) are the solutions to the equation. If the discriminant is positive, the parabola crosses the x-axis at two points. If zero, it touches at one point. If negative, the parabola never crosses the x-axis. The coefficient a determines the width: smaller |a| means a wider parabola, larger |a| means narrower.

Completing the Square Derivation

The quadratic formula is derived from completing the square on ax² + bx + c = 0. Divide by a: x² + (b/a)x + c/a = 0. Move the constant: x² + (b/a)x = -c/a. Add (b/(2a))² to both sides: (x + b/(2a))² = b²/(4a²) - c/a = (b² - 4ac)/(4a²). Take the square root: x + b/(2a) = ±sqrt(b² - 4ac)/(2a). Solve for x: x = (-b ± sqrt(b² - 4ac))/(2a). Understanding this derivation helps you remember the formula and know when to apply it.

Vieta's Formulas: Quick Root Checking

Vieta's formulas provide a quick way to verify your roots without substituting back into the original equation. For ax² + bx + c = 0 with roots r and s: the sum of roots r + s = -b/a, and the product of roots r x s = c/a. For example, in x² - 5x + 6 = 0 with roots 2 and 3: sum = 2 + 3 = 5 = -(-5)/1, and product = 2 x 3 = 6 = 6/1. Both check out. These formulas are especially useful for quickly verifying answers on exams.

Common Mistakes to Avoid

• Forgetting to set the equation to zero: The quadratic formula requires the form ax² + bx + c = 0. If your equation is 3x² = 12x - 5, you must rearrange to 3x² - 12x + 5 = 0 before identifying a = 3, b = -12, c = 5.
• Dropping the negative sign in -b: The formula starts with -b, not b. If b is already negative (say b = -5), then -b = 5 (positive). If b is positive (b = 3), then -b = -3. Sign errors here are the most common mistake.
• Dividing by 2a instead of the entire denominator: The formula divides the entire numerator (-b ± sqrt(D)) by 2a, not just the square root. Both terms in the numerator are divided by 2a. Writing it as x = -b ± sqrt(D)/(2a) is incorrect — use parentheses: x = (-b ± sqrt(D)) / (2a).
• Forgetting that a cannot be zero: If a = 0, the equation bx + c = 0 is linear, not quadratic. The quadratic formula does not apply (division by zero in the denominator). Use simple algebra instead: x = -c/b.
• Not checking for extraneous solutions: When quadratic equations arise from solving other equations (like rational or radical equations), always substitute roots back into the original equation. Some solutions may be extraneous — they satisfy the quadratic but not the original equation.

Frequently Asked Questions