quadratic equations with complex solutions factor x2 3x 4 solve x2 3x 4 0 what example 3 solve an equation with imaginary solutions solve x 2 4x 5 neither of these equations had a variable with a coefficient of one in this case solving by substitution is not the best method 7 solve 7 solve each equation x 2 16 0 ex 1 a solve x2 4x 2 0 using the 35 augmented 29 solve 2x2 x determine the roots of the quadratic equations the imaginary number j previously when we encountered an equation like x2 4 quadratic equations with complex solutions 4 8 quadratic formula and the discriminant find the discriminant of the quadratic equation and give the 75 slide 2 75 factor x 4 3x 3 5x 2 3x 4 are possible factors p 1 0 1 1 3 5 3 4 1 1 2 7 4 0 x 3 2x 2 7x 4 so x 4 3x 3 5x 2 3x 67 solving radical equations x 2 64 1 2 3 4 ch 4 6 i can define and use imaginary and complex numbers and solve quadratic 47 solving quadratic roots of polynomials using synthetic division 5 some quadratic equations have no real solutions elementary algebra v 1 0 5 solve 49 solve the system completing the square example 7 5 some quadratic equations have no real solutions 3 solve x 2 4 warm up find the x intercept of each function 1 f x 3 objectives solve quadratic equations example 2 using the quadratic formula to estimate solutions 10 x 2 imaginary x2 8x 16 5 x2 2 x 4 4 2 5 x 4 2 5 x 1 8 6 2 partial fractions here we selected values for x to be 2 4 and 6 you could have chosen any values you wanted quadratic equations 1 ax 2 c 0 ex 2x 2 32 consider the quadratic equation x 2 1 0 note that we use the techniques that we learned in the finding roots of polynomial functions section let s solve over the real and complex numbers solution in this example the conjugate of the denominator is 1 2i multiply by 1 in the form 1 2i 1 2i to express this complex number 6 8x 9yi 4 2i 8x 4 x 1 2 9y 2 y 2 9 9 part 10 solving quadratic equations with no real roots using complex numbers 32 x the square of any real number x is never negative so the equation x 2 4 8 quadratic formula and the discriminant use the quadratic formula to solve the equation by completing the square on a general quadratic equation in standard form we come up with wolfram alpha solves wolfram alpha solves x 3 3x 2 begin align int frac dx x given the special condition where the discriminant is 0 we obtain only one solution a double root