Derive Quadratic Formula

Derive Quadratic Formula Derive Quadratic Formula A Quadratic equation is a univariate polynomial equation of the second degree. A general quadratic equation can be written in the form Ax2 + bx + c = 0 where x represents a variable or an unknown, and a, b, and c are constants with a ≠0. (If a = 0, the equation is a linear equation.) The constants a, b, and c are called respectively, the quadratic coefficient, the linear coefficient and the constant term or free term. The term "quadratic" comes from quadratus, which is the Latin word for "square". Quadratic equations can be solved by factoring, completing the square, graphing, Newton's method, and using the quadratic formula (given below).

Derivation of Quadratic formula : To solve ax^2 + bx + c = 0 where a ( not 0 ), b, c are constants which can take real number values. ax^2 + bx + c = 0 or ax^2 + bx = -c Dividing by 'a' on both sides, we get x^2 + (b/a)x = -c/a Know More About :- Solving Equations with Exponents

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or x^2 + 2x(b/2a) = -c/a .........(i) The L.H.S. of equation(i) has (first term)^2 and 2(first term)(second term) terms where fist term = x and second term = (b/2a). If we add (second term)^2 {= (b/2a)^2}, the L.H.S. of equation(i) becomes a perfect square. Adding (b/2a)^2 to both sides of equation(i), we get x^2 + 2x(b/2a) + (b/2a)^2 = -c/a + (b/2a)^2 or (x + b/2a)^2 = b^2/4a^2 - c/a = ( b^2 – 4ac)/(4a^2) or (x + b/2a) = ±SquareRoot{( b^2 - 4ac)/(4a^2)} = ±SquareRot( b^2 – 4ac)/2a or x = -b/2a ±SquareRoot(b^2 – 4ac)/2a or x = {-b ±SquareRoot(b^2 - 4ac)}/2a This is the Quadratic Formula. (Derived.) I Applying Quadratic Formula in Finding the roots : Example :- Solve x^2 + x - 42 = 0 using Quadratic Formula.

Comparing this equation with ax^2 + bx + c = 0, we get a = 1, b = 1 and c = -42 x = {-b ±SquareRoot(b^2 - 4ac)}/2a = [ (-1) ±SquareRoot{(1)^2 - 4(1)(-42)}]/2(1) = [ (-1) ±SquareRoot{1 + 168}]/2(1) = [ (-1) ±SquareRoot{169}]/2(1) = [(-1) ± 13]/2(1) = (-1 + 13)/2, (-1 - 13)/2 = 12/2, -14/2 = 6, -7 Ans. Read More About :- Addition and Subtraction of Polynomials

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