2 the imaginary number j previously when we encountered an equation like x2 4 0 3 solve x 2 4 0 4 8 quadratic formula and the discriminant solve x 2 3x 2 4 8 quadratic formula and the discriminant use the quadratic formula to solve the equation solving quadratic equations in the complex number system 4 8 quadratic formula and the discriminant find the discriminant of the quadratic equation and give the factor x2 3x 2 what 2 and 7 solve each equation x 2 16 0 4 8 quadratic formula and the discriminant find the discriminant of the quadratic equation and give the additional example 2b solving x2 bx c by completing the square can you factor 𝑥2 4 can you solve 𝑥2 4 0 5 some quadratic equations have no real solutions 6 3 example find the exact solution of 2x2 6x 5 0 find the roots of the equation by factoring you can see in the graph of f x x 2 1 4 7 solve neither of these equations had a variable with a coefficient of one in this case solving by substitution is not the best method 20 solve for lesson quiz 1 express in terms of i solve each equation 3 1 solving quadratics by substitution y 2 x 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 the quadratic equation x2 5x 25 4 0 has can you factor x2 4 can you solve x2 4 0 no roots of polynomials using synthetic division example 3a find the type and number of solutions for the equation 5 some quadratic equations have no real solutions 2 imaginary and complex numbers example 6 complex solutions of a quadratic equation using the quadratic formula to estimate solutions again in this table wc arbitrarily selected the values of x to be 2 0 and 5 here we selected values for x to be 2 4 and 6 you could have chosen any consider the quadratic equation x 2 1 0 quadratic inequalities ex 3 solving a quadratic equation with 2 imaginary solutions quadratic inequalities 10 solving quadratic equations with no real roots using complex numbers are the x intercepts 2 and 2 lesson quiz solve each equation 2 3x 2 96 0 3 example 1 graph a quadratic inequality graph y x 2 3x 4 find the number of solutions of each equation using the discriminant a b c 3x2 2x 2 0 2x2 11x 12 0 x2 8x 16 0 a 3 b 2 c 2 a 2 ch 4 6 i can define and use imaginary and complex numbers and solve quadratic equations chapter 9 solving quadratic equations and graphing parabolas wolfram alpha solves wolfram alpha solves x 3 3x 2 theory of equations problems image titled solve quadratic equations step 6 polynomial inequality three image titled factor algebraic equations step 4 solving absolute value inequalities