Enter the coefficients of ax² + bx + c = 0 to find its solutions, real or imaginary, with the working shown.
A quadratic equation has the form ax² + bx + c = 0, where a is not zero. The quadratic formula gives its solutions: x = (−b ± √(b² − 4ac)) ÷ 2a. The two signs, plus and minus, give the two solutions. The formula works for every quadratic equation, including ones that cannot be factored neatly, which is why it is taught as the general method.
The part under the square root, b² − 4ac, is called the discriminant, and it tells you what kind of solutions to expect. If it is positive, there are two different real solutions. If it is zero, there is one real solution that counts twice. If it is negative, there are no real solutions, and the two solutions are complex numbers made of a real part and an imaginary part, written with i, where i is the square root of −1.
Type the coefficients a, b and c. Whole numbers, decimals and fractions such as 3/4 all work. The calculator shows the equation, the discriminant, each solution, and the steps from the formula to the answer. Where the coefficients are whole numbers it also simplifies square roots, so a result of 0 ± √8 over 2 is shown as ± √2. If a is zero the equation is linear, and the calculator solves it that way.
For ax² + bx + c = 0, x equals negative b, plus or minus the square root of b squared minus 4ac, all divided by 2a.
The discriminant b² − 4ac shows the kind of solutions. Positive means two real solutions, zero means one repeated real solution, and negative means two complex solutions.
When the discriminant is negative, the square root is of a negative number, which is written using i, the square root of −1. The solutions come as a pair, a + bi and a − bi.
Then the equation is not quadratic. It becomes bx + c = 0, which has the single solution x = −c ÷ b, and the calculator solves it that way.
Yes. Type a fraction such as 3/4 or −1/2 in any of the boxes, and decimals work too.
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