Tell me the derivation of shridharacharya Quadratic formula

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Let's derive the Sridharacharya Quadratic Formula. We start with the standard form of a quadratic equation: a x squared plus b x plus c equals zero, where a is not equal to zero. First, we divide all terms by a to get x squared plus b over a times x plus c over a equals zero. Then, we move the constant term to the right side, giving us x squared plus b over a times x equals negative c over a. Now we complete the square on the left side. To do this, we add the square of half the coefficient of x to both sides. The coefficient of x is b over a, so half of it is b over 2a, and its square is b squared over 4a squared. Adding this to both sides gives us x squared plus b over a times x plus b squared over 4a squared equals negative c over a plus b squared over 4a squared. The left side is now a perfect square, which we can rewrite as: x plus b over 2a, all squared, equals b squared minus 4ac, all over 4a squared. Now we take the square root of both sides. This gives us x plus b over 2a equals plus or minus the square root of b squared minus 4ac, all over 2a. Next, we isolate x by subtracting b over 2a from both sides. This gives us x equals negative b over 2a plus or minus the square root of b squared minus 4ac, all over 2a. Since these terms have a common denominator of 2a, we can combine them to get x equals negative b plus or minus the square root of b squared minus 4ac, all over 2a. This is the famous Sridharacharya Quadratic Formula, which gives us the solutions to any quadratic equation. The discriminant, b squared minus 4ac, tells us about the nature of the roots of a quadratic equation. When the discriminant is positive, the equation has two distinct real roots, as shown by the blue parabola which crosses the x-axis at two points. When the discriminant equals zero, there is exactly one repeated real root, as shown by the green parabola which touches the x-axis at exactly one point. When the discriminant is negative, there are no real roots, only complex conjugate roots, as shown by the red parabola which doesn't intersect the x-axis at all. The Sridharacharya formula works in all these cases, giving us the complete solution set for any quadratic equation. To summarize what we've learned: The Sridharacharya Quadratic Formula provides a universal method for solving any quadratic equation in the form ax squared plus bx plus c equals zero. We derived this formula using the completing the square method, which involves several algebraic steps. The discriminant, b squared minus 4ac, tells us whether the equation has two distinct real roots, one repeated real root, or two complex conjugate roots. The final formula states that x equals negative b plus or minus the square root of b squared minus 4ac, all divided by 2a. This elegant formula was discovered by the Indian mathematician Sridharacharya in the 12th century, predating similar European discoveries by several centuries.