Quadratic Formula Calculator

Solving x Squared: Understanding Quadratic Equations

A quadratic equation is any equation where the highest power of x is 2, which is why it is also called a second-degree equation. They look like ax squared plus bx plus c equals 0, where a, b, and c are numbers. Quadratic equations show up everywhere in the real world: the path of a thrown baseball, the shape of a satellite dish, the profit curve of a business, and the trajectory of a roller coaster are all described by quadratic equations. Solving them means finding the values of x that make the equation true.

Our calculator solves any quadratic equation instantly. Enter the values of a, b, and c and get both solutions, the discriminant, the vertex of the parabola, and a graph. It also shows the step-by-step solution so you can follow along and learn the process.

The quadratic formula: x = (-b plus or minus the square root of (b squared minus 4ac)) / (2a). This single formula solves every quadratic equation that has a solution. For the equation x squared minus 5x plus 6 = 0 (a=1, b=-5, c=6): x = (5 plus or minus the square root of 25 minus 24) / 2 = (5 plus or minus 1) / 2, giving x = 3 and x = 2. The formula works for all quadratics, no matter how messy the numbers get.

The Discriminant: What It Tells You About the Solutions

What is the discriminant? It is the part inside the square root of the quadratic formula: b squared minus 4ac. This single number tells you everything about what kind of solutions to expect before you even finish the calculation. It is like a sneak peek at the answer.

Positive discriminant: If b squared minus 4ac is greater than zero, the equation has two different real solutions. For x squared minus 5x plus 6 = 0: (-5) squared minus 4(1)(6) = 25 minus 24 = 1 (positive), so there are two solutions. This is the most common case and means the parabola crosses the x-axis at two points.

Zero discriminant: If b squared minus 4ac equals zero, the equation has exactly one real solution (technically, the two solutions are the same). For x squared minus 6x plus 9 = 0: (-6) squared minus 4(1)(9) = 36 minus 36 = 0, so the single solution is x = 3. The parabola just touches the x-axis at one point, called the vertex. This happens when the quadratic is a perfect square.

Negative discriminant: If b squared minus 4ac is less than zero, the equation has no real solutions, only complex ones. For x squared plus 1 = 0: 0 minus 4(1)(1) = -4 (negative). The square root of a negative number is not a real number. This means the parabola floats entirely above (or below) the x-axis without ever touching it.

Parabolas: The Shape of Quadratic Equations

Every quadratic equation graphs as a parabola, a smooth U-shaped curve. If a is positive, the parabola opens upward like a smile. If a is negative, it opens downward like a frown. The vertex is the highest or lowest point of the parabola, and it always occurs at x = -b / (2a). This is the turning point where the function changes direction.

Finding the vertex: For y = x squared minus 4x plus 3, the vertex is at x = -(-4) / (2 times 1) = 2. Plug in x = 2: y = 4 minus 8 + 3 = -1. The vertex is at the point (2, -1). Since a is positive, this is the minimum point. The parabola goes down to -1 at x = 2 and then curves back up on both sides.

Axis of symmetry: Every parabola has a vertical line of symmetry that passes through the vertex. For the equation above, the axis of symmetry is the vertical line x = 2. If you fold the graph along this line, the two halves match perfectly. The two solutions (where the parabola crosses the x-axis) are always equally distant from the axis of symmetry.

Quadratic Equations in Real Life

Projectile motion: When you throw a ball, its height over time follows a parabola. A ball thrown upward from 5 feet with an initial velocity of 40 feet per second has a height of h = -16t squared plus 40t plus 5. The vertex tells you the maximum height (30 feet at t = 1.25 seconds). The roots tell you when the ball hits the ground (at t = about 2.62 seconds). Quadratic equations literally describe the flight of every thrown object.

Business profit: A company's profit often follows a quadratic curve. At low prices, profit is low. As the price increases, profit rises. But if the price gets too high, sales drop and profit falls again. The vertex of the profit parabola tells the company the optimal price that maximizes profit. If profit = -2p squared plus 120p minus 500, the maximum profit occurs at p = 30 dollars.

Architecture and design: The arches of bridges, the curves of skateboard ramps, the shape of fountain sprays, and the reflectors in flashlights and satellite dishes are all parabolas. A parabolic reflector focuses incoming parallel rays to a single point (the focus), which is why satellite dishes, telescope mirrors, and car headlights use this shape. The quadratic equation is literally built into the technology we use every day.

When to Use the Formula vs. Factoring

Factoring is faster when it works. If you can spot the factors easily, like x squared minus 5x plus 6 = (x minus 2)(x minus 3), then the solutions are immediately x = 2 and x = 3. Factoring works great for simple equations with small, whole-number solutions. It is the first method to try because it is quick and clean.

The formula always works. For equations like 3x squared minus 7x plus 2 = 0, where the factoring is not obvious, the quadratic formula gives you the answer reliably: x = (7 plus or minus the square root of 49 minus 24) / 6 = (7 plus or minus 5) / 6, giving x = 2 and x = 1/3. The formula handles messy numbers, fractions, and irrational solutions that are hard to factor.

Completing the square is a third method that is useful for rewriting quadratics in vertex form. It is how the quadratic formula itself was derived, and it is used in calculus and physics for more advanced problems. Our calculator uses the formula as the primary method because it is universal and reliable for all quadratic equations.