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XIII SOLVING QUADRATIC EQUATIONS USING THE QUADRATIC FORMULA. A formula for solving quadratic equations can be derived by completing the square for the standard form of a quadratic equation, 0 ax 2 bx c . The solutions of the equation are found by applying the formula : b b 2 4ac x 2a In the formula, a if the coefficient of the quadratic term ( x 2 ) , b is the coefficient of the linear term (x), and c is the constant term. The quadratic formula will always yield the solution to a given quadratic equation, but should be used only when the equation cannot be factored quickly, or when completing the square is not efficient. Example 1. Solve 3x 2 2 x 4 0 using the quadratic formula. In this equation, a = 3, b = 2, and c = -4. Substituting these values into the formula will result in the solutions to the equation. x x x 2 22 4(3)(4) 2(3) 2 4 4 12 2 6 4 48 6 2 52 x 6 2 2 13 x 6 2(1 13) x 2(3) x 1 13 3 The solution set is 1 13 1 13 , 3 3 TO SOLVE A QUADRATIC EQUATION USING THE QUADRATIC FORMULA : 1) 2) 3) 4) Write the equation in standard form, ax 2 bx c 0 Identify the values of a, b and c in this equation Replace these values into the quadratic formula Simplify Example 2. Solve 2t 2 3t 5 0 over the set of complex numbers. 1) The equation is in standard form. 2) a = 2, b = -3, and c = 5 3) Replace these values in the formula : x (3) (3) 2 4 (2)(5) 2(2) 3 9 4 (10) 4 3 9 40 x 4 3 31 x 4 3 i 31 x 4 x The solution set is 3 i 31 3 i 31 , 4 4