Why does the Quadratic Formula have a \pm in it?

The \pm symbol represents two branches of the solution: one using addition and one using subtraction. Most quadratic equations have two solutions (where the parabola crosses the x-axis), and the formula finds both at once. If the discriminant is zero, both branches give the same answer.

What happens if the discriminant is negative?

A negative discriminant means \sqrt{b^2 - 4ac} is the square root of a negative number. In algebra, this means there are no real solutions—the parabola does not cross the x-axis. Mathematically, the formula still works using complex numbers, but for now, you can say no real solutions exist.

Do I have to use the Quadratic Formula, or can I factor instead?

Factoring is faster if the quadratic factors nicely into small integers, but not all quadratics factor cleanly. The Quadratic Formula is the guaranteed, universal method that works for any quadratic equation in standard form, making it the safer choice when factoring seems hard.

What does the discriminant b^2 - 4ac tell me?

The discriminant predicts the nature of the solutions before you finish the calculation: if it is positive, you get two distinct real solutions; if zero, one repeated solution; if negative, no real solutions. It also sits under the square root in the formula, so its value directly determines what you can simplify.

Quadratic Formula

The Quadratic Formula solves any equation of the form ax^2 + bx + c = 0 in one step, giving both solutions at once without factoring.

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

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What each symbol means

What Quadratic Formula takes

Symbol	Meaning
$a$	The coefficient of the $x^2$ term in the equation $ax^2 + bx + c = 0$ ; it must be nonzero (if $a = 0$ , the equation is not quadratic).
$b$	The coefficient of the $x$ term in the equation $ax^2 + bx + c = 0$ ; it can be positive, negative, or zero.
$c$	The constant term in the equation $ax^2 + bx + c = 0$ ; the number with no $x$ attached, which can be positive, negative, or zero.
$x$	The variable you are solving for; the Quadratic Formula gives you the value(s) of $x$ that make the equation true.

When to use it

Use the Quadratic Formula when you need to solve a quadratic equation and factoring is difficult or impossible.

Level

Usually taught in: Algebra I · Appears on: SAT, ACT

Worked examples

1. Solve a quadratic with two integer roots

Problem

Solve

x^2 + 5x + 6 = 0

for

x

$a = 1, \quad b = 5, \quad c = 6$
Identify the coefficients in $x^2 + 5x + 6 = 0$ .
$x = \frac{-5 \pm \sqrt{5^2 - 4(1)(6)}}{2(1)}$
Substitute $a=1$ , $b=5$ , $c=6$ into the Quadratic Formula.
$x = \frac{-5 \pm \sqrt{25-24}}{2} = \frac{-5 \pm \sqrt{1}}{2} = \frac{-5 \pm 1}{2}$
Simplify under the square root: $25 - 24 = 1$ , and $\sqrt{1} = 1$ .
$x = -2 \text{ or } x = -3$
Evaluate both solutions: $\frac{-5+1}{2} = \frac{-4}{2} = -2$ and $\frac{-5-1}{2} = \frac{-6}{2} = -3$ .

Answer: $x = -2 \text{ or } x = -3$

The two integer solutions tell us that the original equation factors nicely as $(x+2)(x+3)=0$ , but the Quadratic Formula works whether or not factoring is obvious.

2. Solve a quadratic with negative and fractional roots

Problem

Solve

2x^2 - 7x + 3 = 0

for

x

$a = 2, \quad b = -7, \quad c = 3$
Identify the coefficients in $2x^2 - 7x + 3 = 0$ ; note that $b$ is negative.
$x = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(2)(3)}}{2(2)} = \frac{7 \pm \sqrt{49 - 24}}{4} = \frac{7 \pm \sqrt{25}}{4}$
Substitute and simplify: $-b = -(-7) = 7$ , and the discriminant is $49 - 24 = 25$ .
$x = \frac{7 \pm 5}{4}$
Evaluate $\sqrt{25} = 5$ .
$x = 3 \text{ or } x = \frac{1}{2}$
The two solutions: $\frac{7+5}{4} = \frac{12}{4} = 3$ and $\frac{7-5}{4} = \frac{2}{4} = \frac{1}{2}$ .

Answer: $x = 3 \text{ or } x = \frac{1}{2}$

This quadratic has one integer root and one fractional root, which would be tedious to find by factoring or completing the square but takes just four steps with the Quadratic Formula.

3. Find break-even points for a profit function

Problem

A school's profit from selling candy boxes is given by

P(x) = -2x^2 + 80x - 600

, where

x

is the number of boxes and

P

is profit in dollars. At what values of

x

is the profit zero? Solve

-2x^2 + 80x - 600 = 0

$a = -2, \quad b = 80, \quad c = -600$
Identify coefficients in $-2x^2 + 80x - 600 = 0$ ; both $a$ and $c$ are negative.
$x = \frac{-80 \pm \sqrt{80^2 - 4(-2)(-600)}}{2(-2)} = \frac{-80 \pm \sqrt{6400 - 4800}}{-4} = \frac{-80 \pm \sqrt{1600}}{-4}$
Substitute into the Quadratic Formula; note that $4ac = 4 \cdot (-2) \cdot (-600) = 4800$ (two negatives make a positive) and $6400 - 4800 = 1600$ .
$x = \frac{-80 \pm 40}{-4}$
Simplify: $\sqrt{1600} = 40$ .
$x = 10 \text{ or } x = 30$
Evaluate both cases: $\frac{-80 + 40}{-4} = \frac{-40}{-4} = 10$ and $\frac{-80 - 40}{-4} = \frac{-120}{-4} = 30$ .

Answer: $x = 10 \text{ or } x = 30$

The school breaks even at both 10 and 30 boxes sold; between these points, profit is positive. The Quadratic Formula handles negative coefficients easily, whereas factoring would be much harder.

Common mistakes

Where Quadratic Formula usually goes wrong

The mistake	Why it is wrong	The fix
Writing $x = -b \pm \sqrt{b^2 - 4ac}$ instead of $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ (forgetting to divide by $2a$ ).	The entire expression $-b \pm \sqrt{b^2 - 4ac}$ must be divided by $2a$ ; leaving out this division gives completely wrong solutions.	Always write the formula as a single fraction: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ with the fraction bar extending under both the $-b$ term and the square root term.
Computing the discriminant incorrectly, especially when $b$ or $c$ is negative, such as calculating $b^2 - 4ac = 25 - 12 = 13$ when you meant $25 - (-12) = 37$ .	A sign error in the discriminant changes the number under the square root, leading to entirely different—and wrong—solutions.	Carefully track signs: calculate $b^2$ first (always positive or zero), then separately compute $4ac$ , paying attention to whether $a$ and $c$ are positive or negative. Remember that $4 \cdot (-2) \cdot (-3) = 24$ (two negatives multiply to positive).
Forgetting the $\pm$ symbol and writing only one solution, such as $x = \frac{-b + \sqrt{b^2 - 4ac}}{2a}$ without the $-$ case.	The $\pm$ means you must evaluate two separate branches (one with $+$ and one with $-$ ); omitting one branch loses half the solutions and is mathematically incomplete.	Always write out both cases: $x = \frac{-b + \sqrt{b^2 - 4ac}}{2a}$ and $x = \frac{-b - \sqrt{b^2 - 4ac}}{2a}$ , or solve them together and list both answers in your final result.

Tips and when to use something else

If the quadratic factors into simple integers (e.g., $(x+2)(x+3)=0$ ), factoring is faster; use Completing the Square or graphing if you need the vertex or want to visualize the parabola.
The discriminant $b^2 - 4ac$ tells you how many real solutions exist: positive gives two different solutions, zero gives one repeated solution, and negative means no real solutions.
When $a = 1$ , the formula simplifies to $x = \frac{-b \pm \sqrt{b^2 - 4c}}{2}$ , which saves arithmetic in the denominator.
Always rearrange the equation to standard form $ax^2 + bx + c = 0$ before identifying $a$ , $b$ , and $c$ ; this prevents sign errors.

Related formulas

DiscriminantThe discriminant, given by b^2 - 4ac, tells you whether a quadratic equation has real solutions and how many without actually solving it.Completing the SquareCompleting the square rewrites a quadratic as a perfect square plus a constant, making it easier to solve and find the parabola's vertex.FOIL MethodThe FOIL Method expands products of two binomials by multiplying each term in the first binomial by each term in the second.Vertex FormVertex Form reveals a parabola's vertex and axis of symmetry at a glance, making it ideal for transformations and optimization problems.

Frequently asked questions

Why does the Quadratic Formula have a $\pm$ in it?: The $\pm$ symbol represents two branches of the solution: one using addition and one using subtraction. Most quadratic equations have two solutions (where the parabola crosses the $x$ -axis), and the formula finds both at once. If the discriminant is zero, both branches give the same answer.
What happens if the discriminant is negative?: A negative discriminant means $\sqrt{b^2 - 4ac}$ is the square root of a negative number. In algebra, this means there are no real solutions—the parabola does not cross the $x$ -axis. Mathematically, the formula still works using complex numbers, but for now, you can say no real solutions exist.
Do I have to use the Quadratic Formula, or can I factor instead?: Factoring is faster if the quadratic factors nicely into small integers, but not all quadratics factor cleanly. The Quadratic Formula is the guaranteed, universal method that works for any quadratic equation in standard form, making it the safer choice when factoring seems hard.
What does the discriminant $b^2 - 4ac$ tell me?: The discriminant predicts the nature of the solutions before you finish the calculation: if it is positive, you get two distinct real solutions; if zero, one repeated solution; if negative, no real solutions. It also sits under the square root in the formula, so its value directly determines what you can simplify.

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Reviewed 2026-09-18