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Online Quadratic Formula Calculator
Solve any quadratic equation ax² + bx + c = 0 using the quadratic formula, with exact and decimal roots, including complex results.
ax2+bx+c=0
x =
-
6
11
±
√19i
11
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What Is This Tool?
The Quadratic Formula Calculator solves equations of the form ax² + bx + c = 0. Enter the three coefficients and it applies the quadratic formula to find the roots. It shows an exact answer using simplified fractions and radicals, along with a decimal approximation, and it handles real, repeated, and complex roots. For example, 11x² + 12x + 5 = 0 gives x = -6/11 ± √19/11 i.
How to Use This Tool?
- Enter the coefficient a (it cannot be zero).
- Enter the coefficients b and c.
- Click Calculate to solve the equation.
- Read the exact answer and its decimal approximation.
Key Features
- Solves any quadratic equation from its coefficients.
- Shows exact roots with simplified fractions and radicals.
- Provides a decimal approximation of each root.
- Handles real, repeated, and complex roots.
- Uses the discriminant to determine the type of roots.
Examples
- x² - 5x + 6 = 0 has roots x = 2 and x = 3.
- x² - 4x + 4 = 0 has one repeated root, x = 2.
- x² + 1 = 0 has complex roots, x = 0 ± i.
- 11x² + 12x + 5 = 0 gives x = -6/11 ± √19/11 i.
Common Use Cases
- Solving quadratic equations in algebra class.
- Finding where a parabola crosses the x-axis.
- Checking homework answers with exact and decimal forms.
- Exploring how the discriminant changes the roots.
- Working with complex roots in math and engineering.
Tips & Best Practices
- Make sure the coefficient a is not zero, or it is not quadratic.
- Enter coefficients exactly, including any minus signs.
- Use whole numbers when you want the exact fraction and radical form.
- Check the discriminant's sign to predict the kind of roots.
- Compare the exact answer with the decimal approximation.
Limitations
- The coefficient a cannot be zero.
- Exact fraction and radical form is shown only for whole-number coefficients.
- Decimal results are rounded for readability.
- It solves quadratic equations only, not higher-degree ones.
Frequently Asked Questions
- What is the discriminant?
- The discriminant is b² - 4ac. It is positive for two real roots, zero for one repeated root, and negative for two complex roots.
- Why can't a be zero?
- If a is zero the equation is no longer quadratic, and the quadratic formula divides by a, which would be undefined.
- What does a complex root look like?
- When the discriminant is negative, the roots include an imaginary part written with i, such as 0 ± i.
- Why are there exact and decimal answers?
- The exact form keeps fractions and radicals precise, while the decimal form gives an easy-to-read approximation.
Key Terminology
- Quadratic equation
- An equation of the form ax² + bx + c = 0 where a is not zero.
- Coefficient
- One of the numbers a, b, or c that multiplies a term in the equation.
- Discriminant
- The value b² - 4ac, which tells you how many and what kind of roots an equation has.
- Root
- A value of x that makes the equation equal to zero.
- Complex root
- A root with an imaginary part, written using i, the square root of -1.