Solving Quadratic Equations

A quadratic equation is any equation that can be written in the form ax² + bx + c = 0, where a ≠ 0. Solving it means finding the values of x that make the equation true. There are three methods: factoring, the square root method, and the quadratic formula. Knowing which one to use — and why — is what separates students who move through this unit confidently from those who get stuck on every problem.

Three Methods at a Glance

1. Factoring — Fastest when it works. Factor the quadratic, set each factor equal to zero, solve. Requires integer factors to exist.

2. Square root method — Use when b = 0 (no middle term). Isolate x², take the square root of both sides, write ±.

3. Quadratic formula — Always works. x = (−b ± √(b²−4ac)) / 2a. Use when factoring fails or when exact form is required.



1) What a Quadratic Equation Is

A quadratic equation contains a variable raised to the second power as its highest degree term. The standard form is ax² + bx + c = 0, where a, b, and c are real numbers and a ≠ 0. The equation must equal zero — this is not optional. Every solution method in this guide requires the equation to be in standard form first.

Some quadratics arrive already in standard form: x² − 5x + 6 = 0. Others require rearranging first: x² = 3x − 2 must become x² − 3x + 2 = 0 before you do anything else. Moving terms to one side and setting the equation equal to zero is the non-negotiable first step regardless of which method you use.

Choosing the right method

The flowchart below shows the decision process. The two questions to ask are: does the equation have a middle term (b = 0 or not), and if so, can the quadratic be factored over the integers? If b = 0, use the square root method. If it factors, use factoring. If neither applies, use the quadratic formula. The quadratic formula always works and is never wrong to use — factoring and the square root method are simply faster when they apply.

Decision flowchart for solving quadratic equations. Step 1: write in standard form ax squared plus bx plus c equals zero. Diamond 1: is b equal to zero? If yes, use the square root method — isolate x squared and take the plus or minus square root. If no, diamond 2: can it be factored? If yes, factor and apply zero product property. If no, use the quadratic formula x equals negative b plus or minus square root of b squared minus 4ac all over 2a. Bottom bar states the quadratic formula always works.
The quadratic formula works for every quadratic equation without exception. Factoring is faster when integer factors exist — but it is a shortcut, not a requirement.

Back to Top



2) Solving by Factoring

Factoring works by rewriting the quadratic as a product of two binomials and then applying the zero product property: if two factors multiply to zero, at least one of them must equal zero. This gives two simple linear equations, each of which has one solution.

For a full breakdown of every factoring method including GCF, the AC method, and special patterns, see the Factoring guide. This section focuses on applying factoring specifically to solve equations.

The zero product property

Once a quadratic is factored into the form (px + q)(rx + s) = 0, set each factor equal to zero separately and solve. The equation must equal zero for this to work — (px + q)(rx + s) = 6 cannot be split into separate equations.

Example: positive c

Solve: x² + 7x + 12 = 0

Factor: (x + 3)(x + 4) = 0
Set each factor equal to zero:
x + 3 = 0 → x = -3
x + 4 = 0 → x = -4

Solutions: x = -3 and x = -4
Verify: (-3)² + 7(-3) + 12 = 9 – 21 + 12 = 0 ✓

Example: negative c (opposite-sign factors)

Solve: x² – x – 6 = 0

Need two numbers that multiply to -6 and add to -1.
(-3) × 2 = -6 and (-3) + 2 = -1 ✓
Factor: (x – 3)(x + 2) = 0
x – 3 = 0 → x = 3
x + 2 = 0 → x = -2

Solutions: x = 3 and x = -2

Example: leading coefficient not 1

Solve: 2x² + 5x – 3 = 0

AC method: a × c = 2 × (-3) = -6
Find pair that multiplies to -6 and adds to 5: (-1) and 6
Rewrite: 2x² – x + 6x – 3
Group: x(2x – 1) + 3(2x – 1) = (2x – 1)(x + 3) = 0
2x – 1 = 0 → x = 1/2
x + 3 = 0 → x = -3

Solutions: x = 1/2 and x = -3

When does factoring not work?

If no pair of integers satisfies both the product and sum conditions, the quadratic does not factor over the integers. This doesn’t mean there are no solutions — it means the solutions are irrational or the quadratic formula is needed. ALEKS and MyMathLab will always tell you which method to use if it’s specified; otherwise the quadratic formula is the safe fallback.

Back to Top



3) The Square Root Method

When a quadratic has no middle term — meaning b = 0 so the equation is ax² + c = 0 — the square root method is the fastest approach. There is no factoring or formula needed. Isolate x² on one side and take the square root of both sides, writing ± to account for both the positive and negative roots.

Example: rational solution

Solve: x² – 25 = 0

Isolate x²: x² = 25
Take square root: x = ±√25
x = ±5

Solutions: x = 5 and x = -5

Example: irrational solution

Solve: 3x² – 18 = 0

Isolate x²: 3x² = 18 → x² = 6
Take square root: x = ±√6

Solutions: x = √6 and x = -√6 (exact form)

The ± symbol is not optional. Dropping it produces only one of the two solutions and is marked wrong on every platform. If the problem asks for exact form, leave the answer as ±√6 rather than converting to a decimal.

✗ Wrong: x² = 6 solved as x = √6 only. Missing the negative solution.

✓ Correct: x = ±√6. Both solutions required.

Back to Top



4) The Quadratic Formula and the Discriminant

The quadratic formula solves any equation in standard form ax² + bx + c = 0 by substituting the values of a, b, and c directly:

x = (-b ± √(b² – 4ac)) / 2a

The expression under the square root — b² − 4ac — is called the discriminant. Calculate it before solving. Its sign tells you exactly how many real solutions to expect and prevents wasted work.

Identifying a, b, and c correctly

The most common formula errors happen before any arithmetic — at the step where a, b, and c are read off the equation. The sign is part of the value. If the equation is x² − 4x + 3 = 0, then b = −4, not 4. The reference card below shows the correct identification for five common equation forms including equations that need rearranging first.

Five-row reference table for identifying a, b, and c in the quadratic formula. Row 1: x squared plus 5x plus 6 equals zero — a equals 1, b equals 5, c equals 6, note that a equals 1 when no coefficient is written. Row 2: x squared minus 4x plus 3 equals zero — a equals 1, b equals negative 4, c equals 3, warning that negative b becomes positive in the formula. Row 3: 2x squared plus 3x minus 5 equals zero — a equals 2, b equals 3, c equals negative 5, note on discriminant calculation. Row 4: 3x squared equals 7x minus 2 in red as not standard form, must rearrange to 3x squared minus 7x plus 2 equals zero first. Row 5: 5x squared minus 45 equals zero — b equals 0 not blank. Bottom bar: the sign is part of the value.
Row 4 is the most important: equations not in standard form must be rearranged before reading off a, b, and c. Reading values from a non-standard equation produces wrong signs and wrong answers.

Three-card discriminant reference chart. Card 1: b squared minus 4ac greater than zero — two distinct real solutions. Parabola crosses x-axis at two points. Example: x squared minus 5x plus 6 equals zero, discriminant equals 1, solutions x equals 2 and x equals 3. Card 2: b squared minus 4ac equals zero — one repeated real solution. Parabola touches x-axis at vertex. Example: x squared minus 6x plus 9, discriminant equals zero, solution x equals 3. Card 3: b squared minus 4ac less than zero — no real solutions, complex roots. Parabola does not cross x-axis. Example: x squared plus x plus 1, discriminant equals negative 3.
A negative discriminant means no real solutions exist — do not attempt to take the square root of a negative number in an algebra course unless complex numbers are explicitly part of the curriculum.

Worked example: two irrational solutions

Solve: x² – 4x – 1 = 0

a = 1, b = -4, c = -1
Discriminant: (-4)² – 4(1)(-1) = 16 + 4 = 20 > 0 → two real solutions

x = (4 ± √20) / 2
Simplify √20 = 2√5
x = (4 ± 2√5) / 2 = 2 ± √5

Solutions: x = 2 + √5 and x = 2 – √5

Worked example: one repeated solution

Solve: 4x² – 12x + 9 = 0

a = 4, b = -12, c = 9
Discriminant: (-12)² – 4(4)(9) = 144 – 144 = 0 → one solution

x = 12 / (2 × 4) = 12/8 = 3/2

Solution: x = 3/2 (double root)

Back to Top



5) Completing the Square

Completing the square is a method for solving quadratic equations that works by rewriting the quadratic in the form (x + p)² = q, then applying the square root method. It is more involved than factoring but always works — and it is the method used to derive the quadratic formula. Some College Algebra and Algebra II courses require it as a standalone technique, particularly on ALEKS and MyMathLab modules that specify the method.

The process

For a quadratic with a = 1, the steps are: move the constant to the right side, take half of b and square it, add that value to both sides, factor the left side as a perfect square trinomial, then apply the square root method.

Solve by completing the square: x² + 6x – 7 = 0

Move constant: x² + 6x = 7
Half of b: 6/2 = 3. Square it: 3² = 9
Add 9 to both sides: x² + 6x + 9 = 7 + 9 = 16
Factor left side: (x + 3)² = 16
Square root both sides: x + 3 = ±4
x = -3 + 4 = 1 or x = -3 – 4 = -7

Solutions: x = 1 and x = -7
Verify: (1)² + 6(1) – 7 = 1 + 6 – 7 = 0 ✓

When a ≠ 1

When the leading coefficient is not 1, divide every term by a first to make the leading coefficient 1, then follow the same steps.

Solve by completing the square: 2x² – 8x – 10 = 0

Divide all terms by 2: x² – 4x – 5 = 0
Move constant: x² – 4x = 5
Half of b: -4/2 = -2. Square it: (-2)² = 4
Add 4 to both sides: x² – 4x + 4 = 5 + 4 = 9
Factor: (x – 2)² = 9
Square root: x – 2 = ±3
x = 2 + 3 = 5 or x = 2 – 3 = -1

Solutions: x = 5 and x = -1

Why does completing the square matter?

The quadratic formula is derived by completing the square on the general form ax² + bx + c = 0. Understanding the method means you can reconstruct the formula from scratch rather than memorizing it. On exams that provide a formula sheet, this doesn’t matter — on exams that don’t, it can be the difference between recovering from a memory lapse and losing all the formula points.



6) Where Students Lose Points

Quadratic equation errors are consistent across every platform. The six below account for the majority of lost points on homework and exams.

Not setting the equation equal to zero first

The zero product property and the quadratic formula both require standard form. Factoring or applying the formula to an equation that hasn’t been rearranged produces wrong answers.

✗ Wrong: Factoring x² + 3x = 10 as x(x + 3) = 10 and writing x = 10 or x + 3 = 10. The equation does not equal zero.

✓ Correct: Rearrange to x² + 3x − 10 = 0, then factor: (x + 5)(x − 2) = 0. Solutions: x = −5 and x = 2.

Sign errors in the quadratic formula

The formula uses −b, not b. When b is negative, −b becomes positive. Substituting b directly without negating it is the single most common formula error.

✗ Wrong: For x² − 4x + 3 = 0, writing x = (−4 ± √(16 − 12)) / 2. b = −4, so −b = +4, not −4.

✓ Correct: x = (4 ± √4) / 2 = (4 ± 2) / 2. Solutions: x = 3 and x = 1.

Forgetting the ± and producing only one solution

Quadratic equations have two solutions (unless the discriminant is zero). Dropping the ± from the quadratic formula or the square root method produces only one solution and is marked wrong on all platforms.

✗ Wrong: x² = 49 solved as x = 7 only.

✓ Correct: x = ±7. Both x = 7 and x = −7 satisfy the equation.

Misreading the discriminant

A negative discriminant means no real solutions — not that a mistake was made. Continuing to solve after finding a negative discriminant (attempting √ of a negative) produces imaginary numbers, which are outside the scope of most College Algebra and Algebra II courses.

✗ Wrong: For x² + x + 1 = 0, computing discriminant = −3 and continuing anyway.

✓ Correct: Discriminant < 0 means no real solutions. State that and stop, unless the problem asks for complex solutions.

Incomplete factoring before applying zero product property

If the GCF is not factored out first, the zero product property may be applied to a partially factored expression, producing only some of the solutions. See the Factoring guide for the complete GCF-first rule.

✗ Wrong: 2x² + 4x = 0 factored as 2x(x + 2) = 0 but only x = −2 recorded as a solution. x = 0 from the factor 2x is also a solution.

✓ Correct: 2x(x + 2) = 0 → x = 0 and x = −2. Both factors produce solutions.

Platform answer entry format

How you enter two solutions varies by platform and costs points when done wrong. MyMathLab typically expects solutions entered as x = −3, x = 4 in separate fields or as a comma-separated list. ALEKS often uses set notation {−3, 4} or asks you to enter each root separately. Hawkes Learning requires exact simplified form — leaving √20 unsimplified instead of 2√5 is marked wrong. Read the entry instructions for each problem before submitting.

Back to Top



7) How FMMC Can Help

Quadratic equations appear in every major algebra platform — MyMathLab, ALEKS, Hawkes Learning, Knewton Alta, Sophia Learning, and WebAssign. If this unit is holding up your grade, FMMC’s algebra experts handle every assignment with an A/B guarantee.

Algebra Homework

Every quadratic equations module handled accurately and on time across all four methods. See our algebra homework help page.

Proctored Exams

Algebra exams on Honorlock and Respondus supported. See our proctored exam page for details.

Full Course Help

Week 1 through final exam across all algebra levels. A or B grade guaranteed — see our guarantee.

Quadratic unit coming up or already behind? Tell us your course, platform, and next due date.

Get Help Now →

Other guides in this series

Factoring  • 
Algebra Homework Help

Back to Top



FAQ: Solving Quadratic Equations

What is completing the square and when do I need it?

Completing the square rewrites a quadratic in the form (x + p)² = q by adding a carefully chosen constant to both sides. It is required when a course or platform explicitly specifies the method — particularly on ALEKS and MyMathLab modules that name the technique. It also underlies the derivation of the quadratic formula. The process: move the constant to the right, take half of b and square it, add to both sides, factor as a perfect square, apply the square root method.

What is a quadratic equation?

A quadratic equation is any equation that can be written in the form ax² + bx + c = 0, where a ≠ 0. It contains a variable squared as its highest degree term. Solving it means finding the value or values of x that make the equation true.

What is the first step in solving any quadratic equation?

Move all terms to one side so the equation equals zero: ax² + bx + c = 0. Every solution method — factoring, square root, and quadratic formula — requires standard form. Skipping this step is the most common setup error.

When should I use the quadratic formula instead of factoring?

Use the quadratic formula when the quadratic does not factor over the integers — meaning no integer pair satisfies both the product and sum conditions. Also use it when a problem explicitly asks for exact form or when the discriminant suggests irrational solutions. The quadratic formula always produces the correct answer regardless of whether factoring was possible.

What is the discriminant and what does it tell you?

The discriminant is b² − 4ac, the expression under the square root in the quadratic formula. If it is positive, the equation has two distinct real solutions. If it equals zero, there is one repeated solution. If it is negative, there are no real solutions — the solutions involve imaginary numbers.

Why does a quadratic equation have two solutions?

Because a quadratic is a degree-2 polynomial, and a polynomial of degree n has exactly n solutions (counting multiplicity) over the complex numbers. For most algebra problems, both solutions are real. When the discriminant is zero, the two solutions happen to be equal (a repeated root). When the discriminant is negative, both solutions are complex.

How do I solve a quadratic with no middle term?

Use the square root method. Isolate x² on one side, then take the square root of both sides and write ±. For example, x² = 49 gives x = ±7. Always write both the positive and negative roots — dropping the ± is marked wrong on every platform.

Can FMMC help with quadratic equation homework and algebra exams?

Yes. FMMC handles algebra homework, quizzes, and proctored exams across MyMathLab, ALEKS, WebAssign, and Hawkes Learning. See our algebra homework help page or get a free quote.

Back to Top

Quadratic Equations Unit Holding Up Your Grade?

Tell us your course, platform, and next due date. FMMC’s algebra experts handle every assignment — A/B guaranteed or your money back.

Get Your Free Quote

Or email: info@finishmymathclass.com  •  A/B Guarantee  •  Algebra Help