Quadratic Equations: The Complete Guide to Solving Them

2026.04.14

Learn.byAuthor

A quadratic equation is any equation that can be written as $ax^2 + bx + c = 0$ . It’s one of the most important topics in algebra and shows up everywhere from the SAT to physics.

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Key Point: Quadratic equations appear on virtually every standardized math test. The SAT typically includes 3~5 questions involving quadratics. Knowing all three solving methods gives you flexibility to pick the fastest approach.

What Is a Quadratic Equation?

Definition and Standard Form

An equation of the form:

ax^2 + bx + c = 0 \quad (a \neq 0)

$a$ : coefficient of $x^2$ (must not be zero)
$b$ : coefficient of $x$
$c$ : constant term

The graph of a quadratic is a parabola — it opens upward when $a > 0$ and downward when $a < 0$ .

Three Solving Methods

Method	Best For	Speed
Factoring	Simple coefficients	Fastest
Completing the Square	Deriving vertex form	Medium
Quadratic Formula	Any quadratic	Reliable

Method 1: Factoring

How It Works

Rewrite the quadratic as a product of two binomials.

x^2 + 5x + 6 = (x + 2)(x + 3) = 0

So $x = -2$ or $x = -3$ .

When to Use It

Factoring works when you can find two numbers that multiply to $c$ and add to $b$ .

Method 2: Completing the Square

Step-by-Step

Convert $ax^2 + bx + c = 0$ into perfect square form.

x^2 + 6x + 5 = 0

(x^2 + 6x + 9) = -5 + 9

(x + 3)^2 = 4 \quad \Rightarrow \quad x = -3 \pm 2

So $x = -1$ or $x = -5$ .

Method 3: The Quadratic Formula

Works for every quadratic equation:

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

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Watch Out: The most common error is mishandling the sign of

b

. When

b

is negative,

-b

becomes positive. Double-check your substitution!

The Discriminant

The expression under the square root, $D = b^2 - 4ac$ , determines the nature of the roots:

$D > 0$ → Two distinct real roots
$D = 0$ → One repeated root (double root)
$D < 0$ → No real roots (two complex roots)

Worked Examples

Example 1: Factoring

Solve $x^2 - 7x + 12 = 0$ .

Show Solution

Find two numbers that multiply to 12 and add to -7: $-3$ and $-4$ .

(x - 3)(x - 4) = 0

$x = 3$ or $x = 4$

Example 2: Quadratic Formula

Solve $2x^2 + 3x - 5 = 0$ .

Show Solution

$a = 2$ , $b = 3$ , $c = -5$

x = \frac{-3 \pm \sqrt{9 + 40}}{4} = \frac{-3 \pm 7}{4}

$x = 1$ or $x = -\frac{5}{2}$

Example 3: Discriminant

How many real roots does $x^2 + 4x + 5 = 0$ have?

Show Solution

$D = 16 - 20 = -4 < 0$

No real roots (two complex roots).

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Pro Tip: On the SAT, try factoring first — it’s faster. If you can’t factor within 15 seconds, switch to the quadratic formula immediately.

Top 3 Common Mistakes

Sign errors in the quadratic formula — especially when $b$ is negative, making $-b$ positive
Forgetting to set the equation to zero — you must have $= 0$ before factoring or using the formula
Dividing by $2a$ , not just $a$ — the denominator is $2a$ , not $2$ or $a$

Quadratic Equations: The Complete Guide to Solving Them

What Is a Quadratic Equation?

Definition and Standard Form

Three Solving Methods

Method 1: Factoring

How It Works

When to Use It

Method 2: Completing the Square

Step-by-Step

Method 3: The Quadratic Formula

The Discriminant

Worked Examples

Example 1: Factoring

Example 2: Quadratic Formula

Example 3: Discriminant

Top 3 Common Mistakes

Related Topics