How to Solve Quadratic Equations: 4 Methods Explained

Q: What is the easiest way to solve quadratic equations?

Factoring is usually the quickest when the numbers are friendly. For x² − 7x + 12 = 0, find two numbers that multiply to 12 and add to −7: −3 and −4. So (x − 3)(x − 4) = 0, giving x = 3 or x = 4. If the equation does not factor easily, the quadratic formula always works, and the square root method is fastest when there is no x term.

Q: What is the formula to solve quadratic equations?

For ax² + bx + c = 0, the quadratic formula is x = (−b ± √(b² − 4ac)) / 2a. Write the equation in standard form first, then identify a, b and c. For x² − 7x + 12 = 0, a = 1, b = −7 and c = 12, so b² − 4ac = 49 − 48 = 1 and x = (7 ± 1)/2, which gives 4 or 3. The value b² − 4ac is called the discriminant.

Q: How do you solve quadratic equations by factoring?

Set the equation equal to zero, factor the quadratic, then set each factor equal to zero. For x² + 5x + 6 = 0, look for two numbers that multiply to 6 and add to 5: 2 and 3. That gives (x + 2)(x + 3) = 0, so x = −2 or x = −3. Always check that one side is zero before factoring, because the zero product rule only works with zero.

Q: How do you solve quadratic equations by completing the square?

Move the constant to the other side, add (b/2)² to both sides, and rewrite the left side as a perfect square. For x² + 6x + 5 = 0, write x² + 6x = −5, then add 9: (x + 3)² = 4. Take square roots to get x + 3 = ±2, so x = −1 or x = −5. If a is not 1, divide every term by a first.

Q: How do you solve quadratic equations using square roots?

Use this method when the equation has no x term. Isolate the squared part, then take the square root of both sides and remember the ± sign. For 2x² = 50, divide by 2 to get x² = 25, so x = 5 or x = −5. It also works for forms like (x − 1)² = 9, which gives x − 1 = ±3, so x = 4 or x = −2.

Q: Is solving quadratic equations hard?

It feels hard at first, but it becomes routine once you know which method to choose. Most mistakes come from skipping steps: forgetting to set the equation equal to zero, dropping the ± sign, or making a sign error with −b in the formula. Practice a few problems with each of the four methods, then check your answers by substituting them back into the original equation.

How to solve quadratic equations: 4 methods explained, step by step: the rule in plain language, solved examples worked one line at a time, and practice questions so you can try it yourself before moving on.

Try it

Equation Solver (Solve for x)

Open the full calculator →

Type the problems from this lesson and check each step.

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

Tutor-style math help

Solve Quadratic Equations: 4 Methods Explained: what to notice and how to work it

Quadratics skill

Quadratic topics connect an equation, a parabola, roots, and a turning point. Read the form first because each form reveals a different feature.

What to notice first

Standard form helps with formulas, factored form shows roots, and vertex form shows the turning point. A good solution starts by using the form you have.

Common student mistake

Do not assume every quadratic has two real x-intercepts. The discriminant tells whether the real graph crosses the x-axis twice, once, or not at all.

Key formulas and cues

$ax^2+bx+c=0$

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

$x=-\frac{b}{2a}$

$y=a(x-h)^2+k$

A reliable path

Read the formFactored, standard, and vertex forms reveal different features.
Choose the methodFactor when friendly, complete the square for structure, or use the formula when needed.
Connect to the graphRoots are x-intercepts and the vertex is the minimum or maximum point.

Worked examples

Factor and solve

Example: $x^2-7x+12=0$

Factor into (x – 3)(x – 4).
Set each factor equal to zero.
Solve both small equations.

Answer: $x=3$ or $x=4$

Find the axis

Example: $y=2x^2-8x+5$

Use x = -b/(2a).
Here a = 2 and b = -8.
Compute 8/4.

Answer: $x=2$

Open pop-up practice

Try one before moving on

Try: Find the axis of symmetry of $y=x^2-6x+2$.

Answer: $x=3$.

Next step: do the matching worksheet or quiz while the method is still fresh, then come back and explain the first step in your own words.

\[ax^2 + bx + c = 0\]

where $a \neq 0$. Quadratics show up everywhere: projectile motion, area problems, profit maximization, and almost every Algebra I final exam.

There are four reliable methods to solve them. This guide walks through each one, when to use it, how to use it, and where students trip up.

Quadratic Equations at a Glance

\[ax^2 + bx + c = 0\]

Solving a quadratic means finding the roots: the x-values where the parabola crosses the x-axis. Those roots are also called zeros or solutions.

First put the equation in standard form, identify $a$, $b$, and $c$, then choose the fastest method.

Which Method Should You Use?

1. Try factoring firstBest when the numbers are friendly, like $x^2-5x+6=0$.

2. Use square rootsBest when there is no middle $bx$ term, like $x^2=49$ or $(x-3)^2=16$.

3. Complete the squareBest when you need vertex form or want to understand how the formula works.

4. Use the formulaAlways works, especially when factoring is not quick or answers are irrational.

Try It: Quadratic Solver and Graph Check

Enter $a$, $b$, and $c$ for $ax^2+bx+c=0$. The tool shows the discriminant, the roots, and a quick graph so you can connect the algebra to the picture.

a b c

Hidden Quadratics: Put It in Standard Form First

Many test questions do not start as $ax^2+bx+c=0$. Move everything to one side before choosing a method.

Original equation	Move to standard form	$a$, $b$, $c$
$x^2 = 3x – 1$	$x^2 – 3x + 1 = 0$	$a=1$, $b=-3$, $c=1$
$2(x^2 – 2x)=5$	$2x^2 – 4x – 5 = 0$	$a=2$, $b=-4$, $c=-5$
$x(x-1)=3$	$x^2 – x – 3 = 0$	$a=1$, $b=-1$, $c=-3$

Quick Check: Try One Before You Scroll

Solve $x^2 – 7x + 12 = 0$

Factor: $(x-3)(x-4)=0$, so $x=3$ or $x=4$.

Solve $x^2 = 81$

Use square roots: $x=\pm 9$.

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

The discriminant is $2^2-4(1)(5)=-16$, so there are no real solutions.

The 4 Methods (and When to Use Each)

Method	Best when..
Factoring	The quadratic factors easily over integers.
Square root property	The equation has no $bx$ term (form $ax^2 + c = 0$).
Completing the square	You want to convert to vertex form, or the leading coefficient is 1 with even $b$.
Quadratic formula	Always works. Use when factoring fails.

Train yourself to ask: "Can this factor? If not, can I use the square root? If not, use the formula."

Method 1: Factoring

Steps: 1. Set the equation to $= 0$. 2. Factor the polynomial. 3. Set each factor equal to 0. 4. Solve each.

How to Solve Quadratic Equations: 4 Methods Explained illustration A

Example 1

Solve $x^2 - 5x + 6 = 0$.

Factor: $(x - 2)(x - 3) = 0$. So $x - 2 = 0$ or $x - 3 = 0$. $x = 2$ or $x = 3$.

Example 2

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

Factor: $(2x - 1)(x + 3) = 0$. $x = \dfrac{1}{2}$ or $x = -3$.

Example 3 (with GCF)

Solve $3x^2 - 12x = 0$.

Factor: $3x(x - 4) = 0$. $x = 0$ or $x = 4$.

The zero product property is the key: if a product equals zero, at least one of the factors must be zero.

Method 2: Square Root Property

Use when the equation looks like $x^2 = k$ or $(x - h)^2 = k$, no $bx$ term.

Rule: $x^2 = k \implies x = \pm\sqrt{k}$.

Example 4

Solve $x^2 = 49$. $x = \pm 7$.

Example 5

Solve $(x - 3)^2 = 16$. $x - 3 = \pm 4$. $x = 3 + 4 = 7$ or $x = 3 - 4 = -1$.

Example 6

Solve $2x^2 - 50 = 0$. $2x^2 = 50$. $x^2 = 25$. $x = \pm 5$.

Don't forget the $\pm$! That is the most common mistake.

Recommended Practice Resources

Method 3: Completing the Square

Use when you want vertex form, or when factoring is awkward.

Steps for $x^2 + bx + c = 0$: 1. Move $c$ to the right side. 2. Take half of $b$, square it, add it to both sides. 3. Write the left side as a perfect square. 4. Square root both sides (with $\pm$). 5. Solve for $x$.

Example 7

Solve $x^2 + 6x - 7 = 0$.

Step 1: $x^2 + 6x = 7$. Step 2: half of 6 is 3; squared is 9. Add 9 to both sides: $x^2 + 6x + 9 = 16$. Step 3: $(x + 3)^2 = 16$. Step 4: $x + 3 = \pm 4$. Step 5: $x = -3 + 4 = 1$ or $x = -3 - 4 = -7$.

Example 8 (leading coefficient ≠ 1)

Solve $2x^2 + 8x - 10 = 0$.

Divide both sides by 2: $x^2 + 4x - 5 = 0$. Move: $x^2 + 4x = 5$. Add $(\frac{4}{2})^2 = 4$: $x^2 + 4x + 4 = 9$. Square: $(x + 2)^2 = 9$. Root: $x + 2 = \pm 3$. $x = 1$ or $x = -5$.

Completing the square is also how you derive the quadratic formula and how you convert standard form to vertex form for graphing.

Method 4: The Quadratic Formula

The universal solver. Works on every quadratic equation, every time.

\[x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

Steps: 1. Put the equation in $ax^2 + bx + c = 0$ form. 2. Identify $a$, $b$, $c$. 3. Plug into the formula. 4. Simplify.

Example 9

Solve $x^2 + 5x + 6 = 0$ using the formula.

$a = 1$, $b = 5$, $c = 6$.

\[x = \dfrac{-5 \pm \sqrt{25 - 24}}{2} = \dfrac{-5 \pm 1}{2}\]

$x = -2$ or $x = -3$. (Same as factoring would give.)

Example 10

Solve $2x^2 - 3x - 1 = 0$.

$a = 2$, $b = -3$, $c = -1$.

\[x = \dfrac{3 \pm \sqrt{9 + 8}}{4} = \dfrac{3 \pm \sqrt{17}}{4}\]

The answers are irrational, exactly the case where you must use the formula.

Example 11 (no real solutions)

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

$a = 1$, $b = 2$, $c = 5$.

\[x = \dfrac{-2 \pm \sqrt{4 - 20}}{2} = \dfrac{-2 \pm \sqrt{-16}}{2}\]

$\sqrt{-16}$ is not a real number. No real solutions. (In complex form: $x = -1 \pm 2i$.)

The Discriminant

Inside the square root sits the discriminant:

How to Solve Quadratic Equations: 4 Methods Explained illustration B

\[D = b^2 - 4ac\]

It tells you how many real solutions the equation has:

$D > 0$: two real solutions.
$D = 0$: one real solution (a repeated root).
$D < 0$: no real solutions (two complex solutions).

Memorize this. It comes up directly on many tests.

Common Mistakes

Forgetting the $\pm$

The square root property always produces two answers (unless the value is 0).

Sign errors with the formula

$-b$ is positive when $b$ is negative. If $b = -3$, then $-b = 3$. Watch carefully.

Forgetting to set $= 0$

You cannot factor or use the formula unless the equation equals zero. Move everything to one side first.

Mishandling fractions

$x = dfrac{4 pm 2}{2}$ is $x = 3$ or $x = 1$, not $x = 6$ or $x = 2$. Divide the whole numerator.

Half of an odd number

If $b = 5$, half of $b$ is $2.5$, and $(2.5)^2 = 6.25$. Use fractions: half is $\dfrac{5}{2}$, squared is $\dfrac{25}{4}$.

Forgetting to simplify the radical

$\sqrt{12} = 2\sqrt{3}$, not just $\sqrt{12}$.

Real-World Example: Projectile Motion

A ball is thrown upward from a 6-foot platform at 32 ft/s. Its height after $t$ seconds is:

$h(t) = -16t^2 + 32t + 6$

When does the ball hit the ground? Set $h(t) = 0$:

$-16t^2 + 32t + 6 = 0$

Using the quadratic formula with $a = -16$, $b = 32$, $c = 6$:

\[t = \dfrac{-32 \pm \sqrt{1024 + 384}}{-32} = \dfrac{-32 \pm \sqrt{1408}}{-32}\]

$\sqrt{1408} \approx 37.52$.

$t \approx \dfrac{-32 - 37.52}{-32} \approx 2.17$ seconds. (Reject the negative time.)

The ball hits the ground after about 2.17 seconds.

Free Resources and Practice Path

Use these next so the idea turns into speed and confidence:

Factoring quadratics lessonPractice the fastest method when the quadratic factors cleanly. Quadratic formula walk-throughUse this when factoring is not quick. Completing the square lessonLearn the bridge to vertex form and the quadratic formula. Completing the square calculatorCheck vertex form and see each step. Quadratic formula worksheetPrintable practice with answers. Math Practice quizzesSearch “quadratic equations,” “factoring,” “quadratic formula,” or “discriminant.”

5-Day Quadratics Practice Path

Day 1: Factor simple quadratics and use the zero-product property.
Day 2: Solve square-root property problems such as $x^2=k$ and $(x-h)^2=k$.
Day 3: Complete the square and connect the result to vertex form.
Day 4: Use the quadratic formula and classify answers with the discriminant.
Day 5: Mix all methods and time yourself: aim for under two minutes per problem.

Frequently Asked Questions

What is the easiest way to solve quadratic equations? Factoring is usually the quickest when the numbers are friendly. For x² − 7x + 12 = 0, find two numbers that multiply to 12 and add to −7: −3 and −4. So (x − 3)(x − 4) = 0, giving x = 3 or x = 4. If the equation does not factor easily, the quadratic formula always works, and the square root method is fastest when there is no x term.

What is the formula to solve quadratic equations? For ax² + bx + c = 0, the quadratic formula is x = (−b ± √(b² − 4ac)) / 2a. Write the equation in standard form first, then identify a, b and c. For x² − 7x + 12 = 0, a = 1, b = −7 and c = 12, so b² − 4ac = 49 − 48 = 1 and x = (7 ± 1)/2, which gives 4 or 3. The value b² − 4ac is called the discriminant.

How do you solve quadratic equations by factoring? Set the equation equal to zero, factor the quadratic, then set each factor equal to zero. For x² + 5x + 6 = 0, look for two numbers that multiply to 6 and add to 5: 2 and 3. That gives (x + 2)(x + 3) = 0, so x = −2 or x = −3. Always check that one side is zero before factoring, because the zero product rule only works with zero.

How do you solve quadratic equations by completing the square? Move the constant to the other side, add (b/2)² to both sides, and rewrite the left side as a perfect square. For x² + 6x + 5 = 0, write x² + 6x = −5, then add 9: (x + 3)² = 4. Take square roots to get x + 3 = ±2, so x = −1 or x = −5. If a is not 1, divide every term by a first.

How do you solve quadratic equations using square roots? Use this method when the equation has no x term. Isolate the squared part, then take the square root of both sides and remember the ± sign. For 2x² = 50, divide by 2 to get x² = 25, so x = 5 or x = −5. It also works for forms like (x − 1)² = 9, which gives x − 1 = ±3, so x = 4 or x = −2.

Is solving quadratic equations hard? It feels hard at first, but it becomes routine once you know which method to choose. Most mistakes come from skipping steps: forgetting to set the equation equal to zero, dropping the ± sign, or making a sign error with −b in the formula. Practice a few problems with each of the four methods, then check your answers by substituting them back into the original equation.

You Have Every Tool You Need

Quadratics reward practice. Drill each method until each takes under two minutes. Then mix them. By exam day, you will look at any quadratic, choose the right method instantly, and solve it without breaking a sweat.

Keep Practicing With the Right Resources

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Original equation	Move to standard form	\(a\), \(b\), \(c\)
\(x^2 = 3x – 1\)	\(x^2 – 3x + 1 = 0\)	\(a=1\), \(b=-3\), \(c=1\)
\(2(x^2 – 2x)=5\)	\(2x^2 – 4x – 5 = 0\)	\(a=2\), \(b=-4\), \(c=-5\)
\(x(x-1)=3\)	\(x^2 – x – 3 = 0\)	\(a=1\), \(b=-1\), \(c=-3\)

How to Solve Quadratic Equations: 4 Methods Explained

Solve Quadratic Equations: 4 Methods Explained: what to notice and how to work it

What to notice first

Common student mistake

Key formulas and cues

A reliable path

Worked examples

Factor and solve

Find the axis

Quadratic Equations at a Glance

Which Method Should You Use?

Try It: Quadratic Solver and Graph Check

Hidden Quadratics: Put It in Standard Form First

Quick Check: Try One Before You Scroll

The 4 Methods (and When to Use Each)

Method 1: Factoring

Example 1

Example 2

Example 3 (with GCF)

Method 2: Square Root Property

Example 4

Example 5

Example 6

Recommended Practice Resources

The Ultimate Algebra Bundle 2027 From Pre-Algebra to Algebra II

Method 3: Completing the Square

Example 7

Example 8 (leading coefficient ≠ 1)

Method 4: The Quadratic Formula

Example 9

Example 10

Example 11 (no real solutions)

The Discriminant

Common Mistakes

Forgetting the \(\pm\)

Sign errors with the formula

Forgetting to set \(= 0\)

Mishandling fractions

Half of an odd number

Forgetting to simplify the radical

Real-World Example: Projectile Motion

Free Resources and Practice Path

5-Day Quadratics Practice Path

Frequently Asked Questions

You Have Every Tool You Need

Keep Practicing With the Right Resources

Effortless Math Team

Related to This Article

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