Solving quadratic equations – AII.4b Quadratic Formula Solving quadratic equations – AII.4b

Discriminant The 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 – 4ac Remember, just type the whole thing into the calculator at once!! Don’t forget the parentheses.

Discriminant Value of the Discriminant Nature of the Solutions Negative 2 imaginary solutions Zero 1 Real – Rational Solution Positive – perfect square 2 Real – Rational Solutions Positive – non-perfect square 2 Real – Irrational Solutions

So how do I find those solutions?? Quadratic formula: 𝑥= −𝑏± (𝑏) 2 −4𝑎𝑐 2𝑎 Wait, does something look familiar? Let’s rewrite it! 𝑥= −𝑏± 𝑑𝑖𝑠𝑐𝑟𝑖𝑚𝑖𝑛𝑎𝑛𝑡 2𝑎 The ‘opposite’ of b. The – just changes the sign of b.

Quadratic Formula – the Steps 𝑥= −𝑏± 𝑑𝑖𝑠𝑐𝑟𝑖𝑚𝑖𝑛𝑎𝑛𝑡 2𝑎 1) find the discriminant 2) plug into the quadratic formula for –b, the discriminant, and a. 3) simplify the radical and denominator 4) simplify the fraction Split the fraction into two if the solutions are rational Just reduce the fraction if the solutions are irrational or imaginary

Let’s look at an example Solve: 3 𝑥 2 +4𝑥=−6 1) discriminant Are we in the correct format? Set the equation equal to zero 3 𝑥 2 +4𝑥+6=0 a = 3, b = 4, c = 6 (𝑏) 2 −4𝑎𝑐⇒ (4) 2 −4 3 6 =−56 Since our discriminant is negative, we have 2 imaginary solutions

Let’s look at an example Solve: 3 𝑥 2 +4𝑥=−6 1) discriminant: −56; 2 imaginary solutions 2) plug into the quadratic formula 𝑥= −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 14 6

Let’s look at an example Solve: 3 𝑥 2 +4𝑥=−6 1) discriminant: −56; 2 imaginary solutions 2) plug into the quadratic formula: 𝑥= −4± −56 2(3) 3) simplify the radical and denominator 4) simplify the fraction Since 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 are x = −4±2𝑖 14 6 = −4±2𝑖 14 6 = −2±𝑖 14 3 x = −2±𝑖 14 3

Solve: 4 𝑗 2 +6=11𝑗 4 𝑗 2 −11𝑗+6=0 a = 4; b = -11; c = 6 1) Discriminant: 4 𝑗 2 −11𝑗+6=0 a = 4; b = -11; c = 6 (b)2 – 4ac => (-11)2 – 4(4)(6) = 25 2 real, rational roots 2) Quadratic Formula 𝑥= −𝑏± 𝑑𝑖𝑠𝑐𝑟𝑖𝑚𝑖𝑛𝑎𝑛𝑡 2𝑎 = −(−11)± 25 2(4)

Solve: 4 𝑗 2 +6=11𝑗 𝑥= −(−11)± 25 2(4) = 11±5 8 1) Discriminant: 25; 2 real, rational roots 2) Quadratic Formula: x = −(−11)± 25 2(4) 3) Simplify the radical and denominator 𝑥= −(−11)± 25 2(4) = 11±5 8 4) Simplify the fraction Since there are rational roots, split the fraction up! 𝑥= 11±5 8 = 11+5 8 and 11−5 8 = 16 8 and 6 8 = 2 and 3 4

Try some on your own…