To solve a quadratic equation means to find all the values of the variable (usually xxx) that make the equation true. These values are called the solutions , roots , or zeros of the quadratic.

What does it mean to solve a quadratic equation?

A quadratic equation is any equation that can be written in the form

ax2+bx+c=0ax^2+bx+c=0ax2+bx+c=0

where aaa, bbb, and ccc are numbers and a≠0a\neq 0a=0.

To solve it means:

  • You are looking for all numbers xxx that, when you plug them in, make the left side equal to zero.
  • Because the degree is 2, there can be up to two solutions (they might be two different numbers, one repeated number, or no real numbers at all).

In simple words:

Solving a quadratic is finding the xxx values that make the equation true.

Mini example

Take the equation:

x2−5x+6=0x^2-5x+6=0x2−5x+6=0

If you try x=2x=2x=2:

22−5⋅2+6=4−10+6=02^2-5\cdot2 +6=4-10+6=022−5⋅2+6=4−10+6=0

So x=2x=2x=2 is a solution. If you try x=3x=3x=3:

32−5⋅3+6=9−15+6=03^2-5\cdot3 +6=9-15+6=032−5⋅3+6=9−15+6=0

So x=3x=3x=3 is also a solution. To “solve” the equation means to find both of these values: x=2x=2x=2 and x=3x=3x=3.

Common ways to solve a quadratic

There are several standard methods you learn in school.

1. Factoring

You rewrite the quadratic as a product of two brackets and then set each bracket to zero.

Example:

x2−5x+6=0⇒(x−2)(x−3)=0x^2-5x+6=0\quad \Rightarrow \quad (x-2)(x-3)=0x2−5x+6=0⇒(x−2)(x−3)=0

So x−2=0x-2=0x−2=0 or x−3=0x-3=0x−3=0, giving x=2x=2x=2 or x=3x=3x=3.

2. Quadratic formula

For any quadratic ax2+bx+c=0ax^2+bx+c=0ax2+bx+c=0, the solutions are given by the quadratic formula :

x=−b±b2−4ac2ax=\frac{-b\pm \sqrt{b^2-4ac}}{2a}x=2a−b±b2−4ac​​

The expression under the square root, b2−4acb^2-4acb2−4ac, is called the discriminant and tells you how many real solutions you get.

3. Completing the square

You rewrite the quadratic into a perfect-square form, like (x−p)2=q(x-p)^2=q(x−p)2=q, and then take square roots.

Example idea (not fully worked):

x2−4x+3=0⇒(x−2)2=1x^2-4x+3=0\quad \Rightarrow \quad (x-2)^2=1x2−4x+3=0⇒(x−2)2=1

From there, take square roots to get the solutions.

4. Graphing

You draw the graph of the quadratic function y=ax2+bx+cy=ax^2+bx+cy=ax2+bx+c.

  • The solutions of the equation ax2+bx+c=0ax^2+bx+c=0ax2+bx+c=0 are exactly the x-intercepts of the graph (where it crosses the x-axis).

So solving the equation is the same as finding where the parabola touches or crosses the x-axis.

How many solutions can a quadratic have?

For real numbers:

  • Two distinct real solutions (parabola crosses the x-axis twice).
  • One real solution (a “double root”, when the parabola just touches the x-axis once).
  • No real solutions (the parabola does not touch the x-axis at all, but there are two complex solutions).

From a more advanced viewpoint, every quadratic has two complex solutions (they may be equal).

Why is this a “trending topic”?

Quadratic equations keep showing up in:

  • High school and college entrance exams (like GCSE, SAT, JEE, etc.), so students search “what does it mean to solve a quadratic equation” a lot.
  • Real-world modeling: physics (projectile motion), economics (profit curves), and many other fields.

You’ll often see students discussing their favorite methods (factoring vs quadratic formula vs “just use a calculator”) on forums and Q&A sites, especially around exam seasons.

Quick multi-view summary

  • Conceptually : Solving a quadratic means finding the roots/solutions/zeros that satisfy the equation.
  • Geometrically : It means finding the x-intercepts of the parabola y=ax2+bx+cy=ax^2+bx+cy=ax2+bx+c.
  • Algebraically : You can do it by factoring, quadratic formula, completing the square, or graphing.

Simple HTML table of methods

[5] [1][9][5] [5] [3][7]
Method Key idea When it’s handy
Factoring Write as (x - r)(x - s) = 0 and solve each factor. When numbers are “nice” and factor easily.
Quadratic formula Use x = [-b ± √(b² - 4ac)] / (2a). Works for every quadratic (very general).
Completing the square Turn into (x - p)² = q and take square roots. Useful for understanding the shape and vertex.
Graphing Plot y = ax² + bx + c and find x-intercepts. Good for visual understanding and estimates.
**TL;DR:** To _solve_ a quadratic equation means to find all values of the variable that make the equation true—its roots—usually using factoring, the quadratic formula, completing the square, or graphing.

Information gathered from public forums or data available on the internet and portrayed here.