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

2. 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.

3. 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

4. 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.

5. 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

6. 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 − =−56
Since our discriminant is negative, we have 2 imaginary solutions

7. 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

8. 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
= −4±2𝑖
= −2±𝑖
x = −2±𝑖

9. 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)± (4)

10. 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)± (4)
3) Simplify the radical and denominator
𝑥= −(−11)± (4) = 11±5 8
4) Simplify the fraction
Since there are rational roots, split the fraction up!
𝑥= 11±5 8 = and 11−5 8 = and 6 8 = 2 and 3 4

11. Try some on your own…