Quadratic Formula Calculator
Apply x = (-b ± √(b²-4ac)) / 2a with detailed steps
Understanding the Quadratic Formula
The quadratic formula is one of the most important formulas in algebra. It provides a direct method to find the roots of any quadratic equation ax² + bx + c = 0, regardless of whether the equation can be factored or not.
The Formula
x = (-b ± √(b² - 4ac)) / 2a
The ± symbol means we get two solutions: one using addition and one using subtraction. These are the two roots of the quadratic equation.
Components of the Formula
-b: The negative of coefficient b from the equation
b² - 4ac: The discriminant, which determines the nature of the roots
2a: Twice the leading coefficient
Why It Works
The quadratic formula is derived by completing the square on the general form ax² + bx + c = 0. This derivation proves that the formula works for all quadratic equations where a ≠ 0.
Frequently Asked Questions
When should I use the quadratic formula?
Use the quadratic formula when you cannot easily factor the equation, or when you need exact decimal solutions. It works for all quadratic equations.
What does the ± symbol mean?
The ± (plus-minus) symbol indicates two separate calculations: one where you add the square root term and one where you subtract it. This gives you both roots of the equation.
What if I get a negative number under the square root?
A negative discriminant (b² - 4ac < 0) means the equation has complex roots. The square root of a negative number involves the imaginary unit i, where i² = -1.
Can I use this for any quadratic equation?
Yes, the quadratic formula works for any quadratic equation in standard form ax² + bx + c = 0 where a ≠ 0. It's the most universal method for solving quadratics.
How do I remember the formula?
Many students use mnemonics or songs. A common one is: "x equals negative b, plus or minus the square root of b squared minus 4ac, all over 2a." Practice using it regularly to memorize it.
What if b or c equals zero?
The formula still works. If b = 0, you get x = ±√(-c/a). If c = 0, one root is always 0 and the other is -b/a. The formula handles all cases.
How accurate are the decimal answers?
The formula gives exact values, but decimal representations may be rounded. For perfect accuracy, keep answers in radical form (with the square root symbol) when the discriminant isn't a perfect square.
Is there a quadratic formula for higher degree equations?
Yes for cubic (degree 3) and quartic (degree 4) equations, but they're much more complex. For degree 5 and higher, there's no general algebraic formula (Abel-Ruffini theorem).