2 DiscriminantThe discriminant tells you what type of roots your equation will have. This can help you decide the best/easiest way to solve it.Quadratic Equation Standard Form: ax2 + bx + c = 0 a, b, and c are coefficients!Discriminant: (b)2 – 4acRemember, just type the whole thing into the calculator at once!! Don’t forget the parentheses.
3 Discriminant Value of the Discriminant Nature of the Solutions Negative2 imaginary solutionsZero1 Real – Rational SolutionPositive – perfect square2 Real – Rational SolutionsPositive – non-perfect square2 Real – Irrational Solutions
4 So how do I find those solutions?? Quadratic formula:𝑥= −𝑏± (𝑏) 2 −4𝑎𝑐 2𝑎Wait, does something look familiar? Let’s rewrite it!𝑥= −𝑏± 𝑑𝑖𝑠𝑐𝑟𝑖𝑚𝑖𝑛𝑎𝑛𝑡 2𝑎The ‘opposite’ of b. The – just changes the sign of b.
5 Quadratic Formula – the Steps 𝑥= −𝑏± 𝑑𝑖𝑠𝑐𝑟𝑖𝑚𝑖𝑛𝑎𝑛𝑡 2𝑎1) find the discriminant2) plug into the quadratic formula for –b, the discriminant, and a.3) simplify the radical and denominator4) simplify the fractionSplit the fraction into two if the solutions are rationalJust reduce the fraction if the solutions are irrational or imaginary
6 Let’s look at an example Solve: 3 𝑥 2 +4𝑥=−61) discriminantAre we in the correct format?Set the equation equal to zero3 𝑥 2 +4𝑥+6=0 a = 3, b = 4, c = 6(𝑏) 2 −4𝑎𝑐⇒ (4) 2 − =−56Since our discriminant is negative, we have 2 imaginary solutions
7 Let’s look at an example Solve: 3 𝑥 2 +4𝑥=−61) discriminant: −56; 2 imaginary solutions2) plug into the quadratic formula𝑥= −𝑏± 𝑑𝑖𝑠𝑐𝑟𝑖𝑚𝑖𝑛𝑎𝑛𝑡 2𝑎 = −4± −56 2(3)3) simplify the radical and denominator−4± −56 2(3) =−4±𝑖 2∙2∙2∙7 6 =−4±2𝑖
8 Let’s look at an example Solve: 3 𝑥 2 +4𝑥=−61) discriminant: −56; 2 imaginary solutions2) plug into the quadratic formula: 𝑥= −4± −56 2(3)3) simplify the radical and denominator4) simplify the fractionSince our solutions are imaginary, there is no need to split it.Can I reduce my coefficients??Yes, divide them by 2!The solutions to 3 𝑥 2 +4𝑥=−6 arex = −4±2𝑖= −4±2𝑖= −2±𝑖x = −2±𝑖