Master quadratic equations through step-by-step interactive lessons. Learn multiple solving methods with real-time practice and instant feedback.
Interactive Learning: Use our Quadratic Equation Calculator alongside this tutorial for hands-on practice.
A quadratic equation is a polynomial equation of degree 2, written in the standard form:
Controls the parabola's width and direction
Affects the parabola's horizontal position
The y-intercept of the parabola
The quadratic formula is the most reliable method for solving any quadratic equation:
Identify coefficients a, b, and c
Make sure the equation is in standard form
Calculate the discriminant (b² - 4ac)
This tells us about the nature of solutions
Apply the quadratic formula
Substitute values and solve for both ± cases
Solve: x² - 5x + 6 = 0
a = 1, b = -5, c = 6
Discriminant = (-5)² - 4(1)(6) = 25 - 24 = 1
x = (5 ± √1) / 2 = (5 ± 1) / 2
Solutions: x = 3 or x = 2
When a quadratic can be factored easily, this method is often faster than the quadratic formula.
ax² + bx + c = a(x - r₁)(x - r₂)
where r₁ and r₂ are the roots
Look for two numbers that multiply to 12 and add to -7
-3 × -4 = 12 and -3 + (-4) = -7 ✓
So: x² - 7x + 12 = (x - 3)(x - 4) = 0
Solutions: x = 3 or x = 4
This method transforms the quadratic into a perfect square trinomial, making it easier to solve.
a(x - h)² + k = 0
where (h, k) is the vertex
Step 1: x² + 6x = -5
Step 2: Take half of 6, square it: (6/2)² = 9
Step 3: x² + 6x + 9 = -5 + 9
Step 4: (x + 3)² = 4
Step 5: x + 3 = ±2, so x = -3 ± 2
Solutions: x = -1 or x = -5