The Quadratic Formula
Understanding the mathematics behind second-degree equations
The Standard Quadratic Formula
For any equation of the form ax² + bx + c = 0
The solutions are given by:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
What is the Quadratic Formula?
The quadratic formula is a method for solving quadratic equations of the form:
ax² + bx + c = 0
where a, b, and c are coefficients, and a ≠ 0.
The formula provides the values of x that satisfy the equation, which correspond to the roots (or zeros) of the quadratic function.
The Discriminant
Δ = b² - 4ac
The expression under the square root is called the discriminant. It determines the nature of the roots:
Positive Discriminant (> 0): Two distinct real roots
Zero Discriminant (= 0): One real root (a repeated solution)
Negative Discriminant (< 0): Two complex roots
Example Problem
Solve the equation: 2x² - 4x - 6 = 0
Here, a = 2, b = -4, c = -6
Substitute into the formula:
\[x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(2)(-6)}}{2(2)}\]
\[x = \frac{4 \pm \sqrt{16 + 48}}{4} = \frac{4 \pm \sqrt{64}}{4} = \frac{4 \pm 8}{4}\]
So the solutions are:
\[x = \frac{12}{4} = 3 \quad \text{and} \quad x = \frac{-4}{4} = -1\]
Role of Coefficients
a - Quadratic Coefficient: Determines the direction and width of the parabola.
b - Linear Coefficient: Influences the position of the vertex and axis of symmetry.
c - Constant Term: Represents the y-intercept of the parabola.
Graphical Representation
The quadratic equation represents a parabola when graphed.
The roots are the x-values where the parabola intersects the x-axis.
The vertex of the parabola is located at:
\[x = \frac{-b}{2a}\]
The quadratic formula gives us the x-intercepts of this parabola.
Interactive Calculator
Enter the coefficients of your quadratic equation to find its roots:
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