Quadratic Formula Calculator
Quadratic Formula Calculator – Solve Quadratic Equations Online
Intro
The Quadratic Formula Calculator online is a free tool that helps you solve quadratic equations quickly and accurately. A quadratic equation is an equation of the form: ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0
where aaa, bbb, and ccc are constants and a≠0a \neq 0a=0.
Solving quadratic equations by hand can be time-consuming and prone to mistakes, especially when dealing with decimals or complex roots. With the Quadratic Formula Calculator online, you can get step-by-step results in seconds—perfect for students, teachers, engineers, and anyone working with algebra or applied mathematics.
Formula / Working
The calculator uses the quadratic formula: x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}x=2a−b±b2−4ac
Where:
- aaa, bbb, and ccc are coefficients of the quadratic equation.
- b2−4acb^2 – 4acb2−4ac is called the discriminant (D).
Cases Based on the Discriminant:
- If D>0D > 0D>0 → two distinct real roots.
- If D=0D = 0D=0 → one real root (repeated).
- If D<0D < 0D<0 → two complex roots.
The calculator applies this formula to give you exact and simplified results.
Step-by-Step Usage
- Enter the coefficients aaa, bbb, and ccc into the calculator.
- Click the Calculate button.
- The calculator instantly displays the solution(s).
- It also shows whether the roots are real or complex.
Examples
Example 1: Quadratic with Two Real Roots
Equation: x2−5x+6=0x^2 – 5x + 6 = 0x2−5x+6=0
Here, a=1a = 1a=1, b=−5b = -5b=−5, c=6c = 6c=6. x=−(−5)±(−5)2−4(1)(6)2(1)x = \frac{-(-5) \pm \sqrt{(-5)^2 – 4(1)(6)}}{2(1)}x=2(1)−(−5)±(−5)2−4(1)(6) =5±25−242=5±12= \frac{5 \pm \sqrt{25 – 24}}{2} = \frac{5 \pm 1}{2}=25±25−24=25±1
✅ Roots: x=2x = 2x=2, x=3x = 3x=3
Example 2: Quadratic with One Repeated Root
Equation: x2−4x+4=0x^2 – 4x + 4 = 0x2−4x+4=0
Here, a=1a = 1a=1, b=−4b = -4b=−4, c=4c = 4c=4. x=−(−4)±(−4)2−4(1)(4)2(1)x = \frac{-(-4) \pm \sqrt{(-4)^2 – 4(1)(4)}}{2(1)}x=2(1)−(−4)±(−4)2−4(1)(4) =4±16−162=42= \frac{4 \pm \sqrt{16 – 16}}{2} = \frac{4}{2}=24±16−16=24
✅ Root: x=2x = 2x=2 (repeated root)
Example 3: Quadratic with Complex Roots
Equation: x2+2x+5=0x^2 + 2x + 5 = 0x2+2x+5=0
Here, a=1a = 1a=1, b=2b = 2b=2, c=5c = 5c=5. x=−2±22−4(1)(5)2(1)=−2±4−202x = \frac{-2 \pm \sqrt{2^2 – 4(1)(5)}}{2(1)} = \frac{-2 \pm \sqrt{4 – 20}}{2}x=2(1)−2±22−4(1)(5)=2−2±4−20 =−2±−162=−2±4i2= \frac{-2 \pm \sqrt{-16}}{2} = \frac{-2 \pm 4i}{2}=2−2±−16=2−2±4i
✅ Roots: x=−1+2i, x=−1−2ix = -1 + 2i, \; x = -1 – 2ix=−1+2i,x=−1−2i
FAQs
Q1. What is a Quadratic Formula Calculator?
It’s an online tool that solves quadratic equations using the quadratic formula.
Q2. Can it solve equations with decimals or fractions?
Yes, the calculator works with all real numbers and provides exact results.
Q3. Will it give complex roots?
Yes, if the discriminant is negative, the calculator shows complex solutions.
Q4. Who can use this tool?
Students, teachers, engineers, and professionals who need quick solutions.
Q5. Is the Quadratic Formula Calculator free?
Yes, it’s 100% free, online, and works on all devices.