Solving Quadratic Equations

Solving a quadratic equation means finding the value or values of $x$ that make it true once it is written in standard form. The real skill is picking the simplest of the three standard methods for the equation in front of you.

Standard Form And Its Symbols

Every quadratic can be written as

with $a \ne 0$. Here $a$, $b$, and $c$ are the coefficients, and the solutions are called the roots. On a graph, the roots are the $x$-values where the parabola meets the $x$-axis.

A quadratic can have two real solutions, one repeated real solution, or no real solutions. Over the complex numbers, every quadratic still has two solutions counting multiplicity.

Why The Discriminant Predicts The Answer

Before doing any algebra, move every term to one side so the other side is $0$. That makes the structure easier to see. From the coefficients you can read the discriminant,

which tells you what kind of real solutions to expect:

• If $b^2 - 4ac > 0$, there are two distinct real solutions.
• If $b^2 - 4ac = 0$, there is one repeated real solution.
• If $b^2 - 4ac < 0$, there are no real solutions.

This comes from the quadratic formula: the discriminant lives under the square root, so its sign controls whether the root is positive, zero, or imaginary. It does not solve the equation by itself, but it tells you what kind of answer should make sense before you start.

The Three Methods And When Each Fits

Factoring works when the quadratic rewrites as a product such as $(x - 2)(x - 3) = 0$. Then the zero-product rule applies: if a product is $0$, at least one factor must be $0$, so $x = 2$ and $x = 3$. It is usually fastest when the expression factors cleanly.

Completing the square rewrites the quadratic in a form like $(x - h)^2 = k$. It is efficient when the equation is close to a perfect-square pattern and factoring is awkward.

The quadratic formula always applies once the equation is in standard form:

It is the most reliable general method, though not always the fastest when the quadratic factors immediately.

Worked Example: Solve $x^2 - 5x + 6 = 0$

This equation is already in standard form, so check whether it factors. You need two numbers that multiply to $6$ and add to $-5$. Those are $-2$ and $-3$, so

Set each factor equal to zero:

giving the solutions

Check both in the original equation:

Both checks work, so the solutions are correct.

Practice: Choose Before You Calculate

Solve $x^2 - 7x + 12 = 0$, but decide the method first. Look for two numbers multiplying to $12$ and adding to $-7$ to test whether it factors cleanly. As a check, the discriminant is $(-7)^2 - 4(1)(12) = 1 > 0$, so expect two distinct real solutions; the answers are $x = 3$ and $x = 4$. If you have time, redo it with the quadratic formula and confirm you get the same pair.

Calculation Pitfalls

The first trap is choosing a method before moving all terms to one side. Solving $x^2 = 5x - 6$ becomes much easier after rewriting it as $x^2 - 5x + 6 = 0$.

The second is dropping one solution. Quadratics can have two real solutions, so after factoring or using the $\pm$, keep both branches when they exist.

The third is using the quadratic formula with the wrong signs for $a$, $b$, or $c$, which usually happens when the equation is not first written in standard form. Quadratics appear in algebra, graphing, optimization, and motion problems, and getting the standard form right is what makes every later step reliable.