How to Solve Quadratic Equations: Formula, Discriminant, Examples

Quadratic equations are one of the most important topics in algebra and show up everywhere from homework to standardized tests. This guide walks through the complete method for solving them: the discriminant, the quadratic formula, a reduced formula for even coefficients, Vieta's formulas, and a couple of shortcuts for spotting roots instantly.

What is a quadratic equation?

A quadratic equation in one variable has the form:

$$a{x}^2+bx+c=0(a\ne 0)$$

where $a$, $b$, and $c$ are known numbers (the coefficients) and $x$ is the unknown. The condition $a\ne 0$ is essential — if $a=0$, the $x^2$ term disappears and the equation becomes linear.

For example, $2x^2-7x+3=0$ is a quadratic equation with $a=2$, $b=-7$, and $c=3$.

The quadratic formula

To solve a quadratic equation, first compute the discriminant (often written as delta):

$$\Delta =b^2-4ac$$

The sign of $\Delta$ tells you how many real solutions the equation has:

When $\Delta \ge 0$, the two roots come from the quadratic formula:

$$x_{1,2}=\frac{-b\pm \sqrt{\Delta }}{2a}$$

The reduced formula

When the coefficient $b$ is even, you can set $b=2b^{\prime }$ and use the reduced discriminant:

$${\Delta }^{\prime }=b^{\prime 2}-ac$$

The roots are then:

$$x_{1,2}=\frac{-b^{\prime }\pm \sqrt{{\Delta }^{\prime }}}{a}$$

This keeps the numbers smaller, so the arithmetic is faster and less error-prone.

Worked examples

Example 1 — Δ > 0. Solve $x^2-5x+6=0$.

Here $a=1$, $b=-5$, $c=6$, so:

$$\Delta =(-5{)}^2-4\cdot 1\cdot 6=25-24=1>0$$

Since $\Delta >0$, the equation has two distinct roots:

$$x_{1,2}=\frac{5\pm \sqrt{1}}{2}\Rightarrow x_1=3,x_2=2$$

Example 2 — using the reduced formula. Solve $x^2-4x+3=0$.

Here $b=-4$ is even, so set $b^{\prime }=-2$. Then:

$${\Delta }^{\prime }=(-2{)}^2-1\cdot 3=4-3=1$$

$$x_{1,2}=\frac{2\pm \sqrt{1}}{1}\Rightarrow x_1=3,x_2=1$$

Example 3 — Δ < 0. Solve $x^2+x+1=0$.

$$\Delta =1^2-4\cdot 1\cdot 1=1-4=-3<0$$

Since $\Delta <0$, the equation has no real solutions.

Vieta's formulas

If the equation $a{x}^2+bx+c=0$ has two roots $x_1$ and $x_2$, then:

$$S=x_1+x_2=-\frac{b}{a}P=x_1\cdot x_2=\frac{c}{a}$$

Vieta's formulas are incredibly useful because they give you the sum and product of the roots without solving the equation. They are commonly used to:

• Guess the two roots quickly when they are nice whole numbers.
• Check your answers after solving.
• Analyze the signs of the roots (if $P<0$ the roots have opposite signs; if $P>0$ and $S>0$ both roots are positive, and so on).

For instance, in $x^2-5x+6=0$ we need two numbers with sum $S=5$ and product $P=6$. Those numbers are $2$ and $3$ — exactly the roots found in Example 1.

Quick shortcuts for special cases

Two special patterns let you read off the roots immediately:

• If $a+b+c=0$, then $x_1=1$ and $x_2=\frac{c}{a}$.
• If $a-b+c=0$, then $x_1=-1$ and $x_2=-\frac{c}{a}$.

For example, $2x^2-5x+3=0$ has $a+b+c=2-5+3=0$, so instantly $x_1=1$ and $x_2=\frac{3}{2}$.

Common mistakes to avoid

• Forgetting the condition $a\ne 0$. Always confirm the leading coefficient is non-zero before applying the quadratic formula.
• Sign errors in $\Delta$. Watch the term $-4ac$ carefully — it flips sign when $c$ is negative.
• Jumping to conclusions when $\Delta <0$. There are no real roots, but over the complex numbers the equation still has two solutions.
• Dropping the $\pm$. When $\Delta >0$ you must take both the $+$ and $-$ signs to get both roots.

Master the discriminant $\Delta$ and Vieta's formulas, and you can solve almost any quadratic equation quickly. Try the examples above in the calculator to check each step of your work.