Solve quadratic equations using the quadratic formula. Get step-by-step solutions for real and complex roots. Perfect for students, teachers, and math professionals.
For the equation: ax² + bx + c = 0
Where: b² - 4ac is the discriminant
Enter the coefficients above to solve the quadratic equation
Scenario: A ball is thrown upward from a height of 2 meters with an initial velocity of 10 m/s. The height h(t) after t seconds is given by h(t) = -4.9t² + 10t + 2. To find when the ball hits the ground, we solve -4.9t² + 10t + 2 = 0. Using this calculator with a = -4.9, b = 10, c = 2, we get t ≈ 2.24 seconds (the positive root, since time cannot be negative). This helps in physics problems involving projectile motion.
A quadratic equation is a polynomial equation of degree 2, written in the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. The highest power of the variable x is 2.
The quadratic formula is x = (-b ± √(b² - 4ac)) / (2a), which gives the solutions to any quadratic equation ax² + bx + c = 0. The ± symbol means we get two solutions.
The discriminant is b² - 4ac. It determines the nature of the roots: if positive, there are two real roots; if zero, there's one real root (repeated); if negative, there are two complex conjugate roots.
A quadratic equation always has exactly two solutions (roots), which may be real or complex. If the discriminant is zero, both solutions are the same (repeated root).
Complex roots occur when the discriminant is negative. They come in conjugate pairs: if a + bi is a root, then a - bi is also a root, where i is the imaginary unit (√-1).
Quadratic equations have real roots when the discriminant (b² - 4ac) is greater than or equal to zero. If the discriminant is exactly zero, there's one real root; if positive, there are two distinct real roots.
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