The Quadratic Formula and the Discriminant

The Quadratic Formula

Introduction to the Quadratic Formula

The quadratic formula provides a universal method for solving any quadratic equation in standard form. A quadratic equation is any equation that can be written as ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0.

• Quadratic Formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

This formula gives the solutions (roots) of any quadratic equation.

Methods for Solving Quadratic Equations

There are several methods to solve quadratic equations. The quadratic formula is especially useful when factoring is difficult or impossible.

Factoring	Sq. Root Property	Complete the Square	Quadratic Formula
Try to factor and set each factor to zero.	Isolate $x^2$ and take the square root of both sides.	Make a perfect square trinomial, then solve for $x$.	Use when factoring is not possible or practical.
Works best when factors are integers.	Works only if there is no $x$ term.	Works for all quadratics, but can be lengthy.	Works for all quadratics.

Example: Solving with the Quadratic Formula

• Example 1: Solve $2x^2 + 3x - 2 = 0$

$a = 2,\ b = 3,\ c = -2$ $x = \frac{-3 \pm \sqrt{3^2 - 4(2)(-2)}}{2 \cdot 2}$ $x = \frac{-3 \pm \sqrt{9 + 16}}{4}$ $x = \frac{-3 \pm 5}{4}$ $x = \frac{2}{4} = \frac{1}{2}$ or $x = \frac{-8}{4} = -2$

• Example 2: Solve $x^2 - 6x + 8 = 0$

$a = 1,\ b = -6,\ c = 8$ $x = \frac{6 \pm \sqrt{(-6)^2 - 4(1)(8)}}{2 \cdot 1}$ $x = \frac{6 \pm \sqrt{36 - 32}}{2}$ $x = \frac{6 \pm 2}{2}$ $x = 4$ or $x = 2$

Practice Problems

• Solve $3x^2 + 4x + 1 = 0$ using the quadratic formula.

• Solve $2x^2 - 5x = -3$ using the quadratic formula.

The Discriminant

Understanding the Discriminant

The discriminant is the expression under the square root in the quadratic formula: $b^2 - 4ac$. It determines the number and type of solutions to a quadratic equation without actually solving it.

• Discriminant Formula:

$D = b^2 - 4ac$

• If $D > 0$: Two distinct real solutions

• If $D = 0$: One real solution (a repeated root)

• If $D < 0$: Two complex (imaginary) solutions

Example: Determining the Number and Type of Solutions

• Example A: $2x^2 + 3x - 2 = 0$

$D = 3^2 - 4(2)(-2) = 9 + 16 = 25 > 0$ Two real solutions

• Example B: $4x^2 - 4x + 1 = 0$

$D = (-4)^2 - 4(4)(1) = 16 - 16 = 0$ One real solution

• Example C: $-10x^2 + 15x - 6 = 0$

$D = 15^2 - 4(-10)(-6) = 225 - 240 = -15 < 0$ Two complex solutions

Practice Problems

• Determine the number and type of solutions for $x^2 + 6x + 10 = 0$.

• Determine the number and type of solutions for $4x^2 + 5 = 0$.

Additional info: The quadratic formula and discriminant are foundational tools in algebra and precalculus, providing a systematic approach to solving and analyzing quadratic equations. Mastery of these concepts is essential for further study in mathematics.