Quadratic Studio

Solve quadratic equations, visualize parabolas, and master the formula.

Standard Form

ax² + bx + c = 0

The standard form of a quadratic equation. 'a', 'b', and 'c' are known values. 'a' cannot be 0.

Quadratic Formula

x = -b ± √(b² - 4ac)
2a

Used to find the roots (solutions) of any quadratic equation.

Discriminant (Δ)

Δ = b² - 4ac
  • Positive: 2 Real Roots
  • Zero: 1 Real Root
  • Negative: 2 Complex Roots

Vertex Form

y = a(x - h)² + k
Vertex is at (h, k)
h = -b / 2a

Mastering the Quadratic Equation

The quadratic equation, ax² + bx + c = 0, is one of the most powerful tools in algebra. It describes everything from the flight path of a kicked soccer ball to the architectural curve of a suspension bridge. While factoring works for simple cases, the Quadratic Formula is the universal key that unlocks the solution to any quadratic equation.

This Quadratic Studio solves the equation for you, showing every step of the work, and visualizes the resulting parabola so you can see exactly what the math represents.

What is the Discriminant?

The term under the square root, b² - 4ac, is called the Discriminant (Δ). It acts as a "scout" that tells you the nature of the roots before you even finish solving.

  • Positive (Δ > 0)The parabola crosses the x-axis twice. There are two distinct real solutions.
  • Zero (Δ = 0)The parabola just touches the x-axis at one point (the vertex). There is exactly one real solution.
  • Negative (Δ < 0)The parabola never touches the x-axis (it floats above or below). The solutions involve imaginary numbers (i).

Quadratic in the Real World

The most famous application of quadratic equations is Projectile Motion in physics. Because gravity accelerates objects downwards at a constant rate, the position of any falling or thrown object follows a quadratic path (a parabola).

Equation: h(t) = -16t² + vt + s

This models height (h) over time (t) in feet.

  • -16t²: The effect of gravity (half of 32 ft/s²).
  • vt: The initial upward velocity.
  • s: The starting height.

Solving this equation for h = 0 tells you exactly when the object will hit the ground.

Vertex Form vs Standard Form

While standard form (ax² + bx + c) is great for the quadratic formula, it's not easy to graph.Vertex Form is simpler for understanding the shape:

y = a(x - h)² + k

Here, `(h, k)` is the exact peak or valley of the parabola.
• If a is positive, the parabola opens Up (smiley).
• If a is negative, it opens Down (frowny).

What about "Completing the Square"?

Completing the square is the method used to derive the quadratic formula. It involves rearranging the equation to create a perfect square trinomial (like `(x+3)²`).

While you can solve equations this way, it is often tedious if `b` is an odd number or a fraction. The Quadratic Formula is essentially the "pre-packaged" version of completing the square—it does the hard algebraic manipulation once, so you don't have to do it every time.

Frequently Asked Questions

Why is there a ± (plus-minus) in the formula?

The ± exists because a square root technically has two answers. Both (2)² and (-2)² equal 4. Therefore, when we take the square root to solve for x, we must account for both the positive and negative direction from the axis of symmetry. This corresponds to the two points where the parabola crosses the x-axis.

What does it mean if the discriminant is negative?

If b² - 4ac is negative, you are trying to take the square root of a negative number. In the real number system, this is impossible, which means the parabola never touches the x-axis. However, in the complex number system, we use 'i' (the imaginary unit) to represent √-1, resulting in two complex solutions.

Can I use the quadratic formula for simple equations?

Yes, the quadratic formula works for ALL quadratic equations, even simple ones like x² - 9 = 0 (where b=0). However, for simple integer roots, Factoring is often faster. For example, x² + 5x + 6 = 0 factors easily to (x+2)(x+3) = 0.

How do I find the Axis of Symmetry?

The Axis of Symmetry is the vertical line that splits the parabola perfectly in half. Its equation is simply x = -b / 2a. This is actually the first part of the quadratic formula, representing the center point before you add/subtract the square root distance.

What happens if 'a' is zero?

If a = 0, the term ax² disappears, and you are left with bx + c = 0. This is no longer a quadratic equation; it is a Linear Equation (a straight line). The quadratic formula requires 'a' to be non-zero.

What is the relationship between roots and factors?

If the solutions (roots) to a quadratic equation are r1 and r2, then the equation can be written in factored form as a(x - r1)(x - r2) = 0. For example, if the roots are 2 and 3, the equation is (x-2)(x-3) = x² - 5x + 6.

What does the 'c' value represent on the graph?

The 'c' value is the Y-intercept. It is the point where the parabola crosses the vertical y-axis (where x = 0). If you plug x=0 into the equation ax² + bx + c, the first two terms become zero, leaving only y = c.

How do I know if the vertex is a minimum or maximum?

Look at the sign of 'a'. If 'a' is positive, the parabola opens upwards (like a cup), so the vertex is the Minimum point. If 'a' is negative, the parabola opens downwards (like a frown), so the vertex is the Maximum point.

Why do we divide by 2a?

Mathematically, the 2a comes from the process of completing the square. Geometrically, it scales the result inversely with the steepness of the parabola. A larger 'a' value (steeper parabola) means the roots are closer together, so dividing by a larger number shrinks the distance.

What are 'Complex Roots'?

Complex roots occur when the parabola doesn't touch the x-axis. They are written in the form a + bi, where 'i' is the square root of -1. They come in conjugate pairs (e.g., 2+3i and 2-3i). While they don't represent physical x-intercepts, they are crucial in advanced engineering and electrical circuit theory.