How to Solve Quadratic Equations: Complete Step-by-Step Methods

A quadratic equation is a polynomial equation of the second degree, meaning it contains a term with a variable raised to the power of 2. The standard form is:

$$ax^2 + bx + c = 0$$

where $a$, $b$, and $c$ are constants, and $a \neq 0$. Solving a quadratic equation means finding the values of $x$ that make the equation true. These values are often called the roots or solutions. Here are the three primary methods to solve them.

Method 1: Factoring

Factoring involves rewriting the quadratic expression as a product of two linear factors. This method is fast but only works when the equation can be factored easily.

When to Use It: Best for simple quadratics where the roots are integers.

Step-by-Step Example: Solve $x^2 + 5x + 6 = 0$.

1. Find two numbers that multiply to $c$ (6) and add up to $b$ (5).

   • The pairs of numbers that multiply to 6 are (1, 6), (-1, -6), (2, 3), and (-2, -3).
   • The pair that adds up to 5 is (2, 3).

2. Rewrite the equation using these numbers. $(x + 2)(x + 3) = 0$

3. Set each factor to zero to find the roots.

   • If $x + 2 = 0$, then $x = -2$.
   • If $x + 3 = 0$, then $x = -3$.

Answer: The solutions are $x = -2$ and $x = -3$.

Method 2: The Quadratic Formula

The quadratic formula is a universal tool that can solve any quadratic equation. It might take longer than factoring, but it always works.

When to Use It: Always. Especially useful when the equation doesn't factor easily or has irrational or complex roots.

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

Step-by-Step Example: Solve $2x^2 - 8x + 3 = 0$.

1. Identify the coefficients $a$, $b$, and $c$.

   • $a = 2$
   • $b = -8$
   • $c = 3$

2. Substitute these values into the quadratic formula. $x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4(2)(3)}}{2(2)}$

3. Simplify the expression. $x = \frac{8 \pm \sqrt{64 - 24}}{4}$ $x = \frac{8 \pm \sqrt{40}}{4}$ $x = \frac{8 \pm 2\sqrt{10}}{4}$

4. Simplify further to get the final roots. $x = \frac{4 \pm \sqrt{10}}{2}$

Answer: The solutions are $x = 2 + \frac{\sqrt{10}}{2}$ and $x = 2 - \frac{\sqrt{10}}{2}$.

Understanding the Discriminant

The part of the quadratic formula inside the square root, $b^2 - 4ac$, is called the discriminant. It tells you about the nature of the roots without fully solving the equation:

• If $b^2 - 4ac > 0$, there are two distinct real roots.
• If $b^2 - 4ac = 0$, there is exactly one real root (a "repeated" root).
• If $b^2 - 4ac < 0$, there are two complex roots (involving imaginary numbers).

Method 3: Completing the Square

Completing the square is a process that converts a quadratic equation into a "perfect square" trinomial, which is easy to solve. This method is also the basis for deriving the quadratic formula.

When to Use It: Useful for converting a quadratic to vertex form, $y = a(x-h)^2 + k$, and when the leading coefficient $a$ is 1 and the middle coefficient $b$ is an even number.

Step-by-Step Example: Solve $x^2 + 6x - 5 = 0$.

1. Move the constant term to the other side. $x^2 + 6x = 5$

2. Take half of the coefficient of $x$, square it, and add it to both sides.

   • Half of 6 is 3.
   • $3^2 = 9$. $x^2 + 6x + 9 = 5 + 9$

3. Factor the left side as a perfect square. $(x + 3)^2 = 14$

4. Take the square root of both sides. $x + 3 = \pm\sqrt{14}$

5. Solve for $x$. $x = -3 \pm\sqrt{14}$

Answer: The solutions are $x = -3 + \sqrt{14}$ and $x = -3 - \sqrt{14}$.

Which Method Should You Use?

• First, try factoring. It's the quickest method if it works.
• If factoring fails or looks difficult, use the quadratic formula. It's reliable and straightforward.
• Use completing the square when you need to find the vertex of a parabola or if your instructor requires it.

By mastering these three methods, you can confidently solve any quadratic equation you encounter.