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Quadratic Equation Solver

Solve quadratic equations effortlessly with our step-by-step calculator. Get instant solutions and detailed explanations for equations in the form ax2+bx+c=0ax^2 + bx + c = 0.

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Master Quadratic Equations with Our Advanced Solver

Our quadratic equation calculator is designed to help students, teachers, and professionals solve mathematical problems efficiently. Whether you're working on algebra homework, preparing forSAT exams, or tackling university mathematics courses, this tool provides comprehensive step-by-step solutions that enhance your understanding of quadratic equations.

The algebra solver online handles equations in the standard form ax2+bx+c=0ax^2 + bx + c = 0, where aa, bb, and cc are real numbers with a≠0a \neq 0. This equation simplifier can process various formats including fractions, decimals, and equations that need to be rearranged. Our math solver is particularly useful for SAT preparation, where time management and accuracy are crucial.

Perfect for high school algebra students learning the fundamentals, college studentsin calculus prerequisites, and professionals who need quick mathematical solutions. Thequadratic equation solver provides not just answers, but detailed explanations that help you understand the underlying concepts and improve your problem-solving skills.

Types of Quadratic Equations

Type of Quadratic EquationExampleSolutionDifficulty Level
Standard Form

x2+5x+6=0x^2 + 5x + 6 = 0

x=−2,−3x = -2, -3

Beginner
Perfect Square

x2+6x+9=0x^2 + 6x + 9 = 0

x=−3x = -3 (double root)

Beginner
With Fractions

2x2−12x−1=02x^2 - \frac{1}{2}x - 1 = 0

x=1,−12x = 1, -\frac{1}{2}

Intermediate
Complex Roots

x2+2x+5=0x^2 + 2x + 5 = 0

x=−1±2ix = -1 \pm 2i

Advanced
Leading Coefficient ≠ 1

3x2−7x+2=03x^2 - 7x + 2 = 0

x=2,13x = 2, \frac{1}{3}

Intermediate

Common Mistakes to Avoid

Forgetting to Check a ≠ 0

If a=0a = 0, the equation becomes linear, not quadratic. Always ensure a≠0a \neq 0 before applying the quadratic formula.

Sign Errors in Formula

In the quadratic formula x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2-4ac}}{2a}, remember the negative sign before bb and the ±\pm symbol.

Ignoring Complex Solutions

When the discriminant b2−4ac<0b^2 - 4ac < 0, the equation has complex solutions. Don't assume there are no solutions.

How to Solve Quadratic Equations

A quadratic equation is an equation where the highest power of the variable is 2. The general form is:

ax2+bx+c=0ax^2 + bx + c = 0

where aa, bb, and cc are constants with a≠0a \neq 0, and xx is the variable we need to solve for.

Step-by-Step Method

1

Check the form

Ensure the equation is in standard form ax² + bx + c = 0

2

Calculate discriminant

Find Δ = b² - 4ac to determine the nature of solutions

3

Apply quadratic formula

Use x = (-b ± √Δ) / (2a) to find solutions

4

Simplify and check

Simplify the results and verify by substitution

Key Formula

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Examples

Simple Quadratic

x2+5x+6=0x^2 + 5x + 6 = 0

Solution:

  1. Identify coefficients: a=1a = 1, b=5b = 5, c=6c = 6

  2. Calculate discriminant: Δ=25−24=1\Delta = 25 - 24 = 1

  3. Apply formula: x=−5±12=−2,−3x = \frac{-5 \pm \sqrt{1}}{2} = -2, -3

x=−2,x=−3x = -2, x = -3

Perfect Square

x2+6x+9=0x^2 + 6x + 9 = 0

Solution:

  1. Identify coefficients: a=1a = 1, b=6b = 6, c=9c = 9

  2. Calculate discriminant: Δ=36−36=0\Delta = 36 - 36 = 0

  3. Apply formula: x=−6±02=−3x = \frac{-6 \pm \sqrt{0}}{2} = -3

x=−3(doubleroot)x = -3 (double root)

Complex Roots

x2+2x+5=0x^2 + 2x + 5 = 0

Solution:

  1. Identify coefficients: a=1a = 1, b=2b = 2, c=5c = 5

  2. Calculate discriminant: Δ=4−20=−16\Delta = 4 - 20 = -16

  3. Apply formula: x=−2±−162=−1±2ix = \frac{-2 \pm \sqrt{-16}}{2} = -1 \pm 2i

x=−1+2i,x=−1−2ix = -1 + 2i, x = -1 - 2i

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Frequently Asked Questions

What is a quadratic equation?

A quadratic equation is an equation where the highest power of the variable is 2. It forms a parabola when graphed and can have 0, 1, or 2 real solutions.

Can this solver handle complex numbers?

Yes, our quadratic equation solver can handle complex solutions. When the discriminant is negative, the calculator will show complex solutions in the form a ± bi.

What if the equation has no real solutions?

If the discriminant b2−4ac<0b^2 - 4ac < 0, the equation has no real solutions but has two complex conjugate solutions.

Can I solve equations that are not in standard form?

Yes, our calculator can handle equations that need to be rearranged. Simply enter the equation as it appears and the solver will help you convert it to standard form.

Is this calculator free to use?

Yes, our quadratic equation solver is completely free to use with no limitations. You can solve as many equations as you need.

How accurate are the solutions?

Our calculator provides highly accurate solutions with step-by-step explanations. It handles both exact fractions and decimal approximations.

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Last updated: 24/08/2025 — Written by the AskMathAI team