Quadratic Formula#
Quadratic Formula#
Theory#
The quadratic formula is a universal method for solving any quadratic equation of the form \(ax^2 + bx + c = 0\) where \(a \neq 0\). It provides a direct way to find the solutions without factoring or completing the square.
The quadratic formula is:
This formula was derived by completing the square on the general quadratic equation. The expression under the square root, \(b^2 - 4ac\), is called the discriminant and is denoted by \(\Delta\) (delta).
The discriminant tells us about the nature of the solutions:
If \(\Delta > 0\): Two distinct real solutions
If \(\Delta = 0\): One repeated real solution (also called a double root)
If \(\Delta < 0\): Two complex conjugate solutions (no real solutions)
Note
The quadratic formula always works for any quadratic equation, making it a reliable fallback when other methods are difficult to apply.
Interactive Visualization: Quadratic Formula Explorer#
Application#
Examples#
Example 1#
Solve using the quadratic formula: \(x^2 + 5x + 6 = 0\)
Method 1: Quadratic Formula
\(x^2 + 5x + 6 = 0 \quad \text{(Identify: a = 1, b = 5, c = 6)}\)
\(\Delta = b^2 - 4ac = 25 - 24 = 1 \quad \text{(Calculate discriminant)}\)
\(x = \frac{-5 \pm \sqrt{1}}{2(1)} = \frac{-5 \pm 1}{2} \quad \text{(Apply formula)}\)
\(x = \frac{-5 + 1}{2} = -2 \text{ or } x = \frac{-5 - 1}{2} = -3 \quad \text{(Calculate both solutions)}\)
Example 2#
Solve: \(2x^2 - 4x - 3 = 0\)
Method 1: Quadratic Formula
\(2x^2 - 4x - 3 = 0 \quad \text{(Identify: a = 2, b = -4, c = -3)}\)
\(\Delta = (-4)^2 - 4(2)(-3) = 16 + 24 = 40 \quad \text{(Calculate discriminant)}\)
\(x = \frac{4 \pm \sqrt{40}}{4} = \frac{4 \pm 2\sqrt{10}}{4} \quad \text{(Apply formula and simplify)}\)
\(x = \frac{2 \pm \sqrt{10}}{2} \quad \text{(Reduce fraction)}\)
Example 3#
Solve: \(x^2 - 6x + 9 = 0\)
Method 1: Quadratic Formula
\(x^2 - 6x + 9 = 0 \quad \text{(Identify: a = 1, b = -6, c = 9)}\)
\(\Delta = (-6)^2 - 4(1)(9) = 36 - 36 = 0 \quad \text{(Discriminant is zero)}\)
\(x = \frac{6 \pm \sqrt{0}}{2} = \frac{6}{2} = 3 \quad \text{(One repeated solution)}\)
Method 2: Recognition of Perfect Square
\(x^2 - 6x + 9 = (x - 3)^2 = 0 \quad \text{(Recognize perfect square)}\)
\(x = 3 \quad \text{(Double root)}\)
Multiple Choice Questions#
Sector Specific Questions: Quadratic Formula Applications#
Key Takeaways#
Important
The quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) solves any quadratic equation
The discriminant \(\Delta = b^2 - 4ac\) determines the nature of solutions
When \(\Delta > 0\): two distinct real solutions
When \(\Delta = 0\): one repeated real solution
When \(\Delta < 0\): two complex conjugate solutions
The quadratic formula is derived by completing the square on \(ax^2 + bx + c = 0\)
Always simplify the final answer by reducing fractions and simplifying radicals
The quadratic formula works even when factoring is difficult or impossible