Section 4.2: Solving Quadratics Using the Quadratic Formula
February 17, 2016
Section 4.2: Solving Quadratic Equations
Part 2: Quadratic Formula
How to derive the quadratic formula by
completing the square...(this is just for fun)
Pre­Calculus
Section 4.2: Solving Quadratics Using the Quadratic Formula
February 17, 2016
If you complete the square using only variables, this is what you get when you solve for x
http://www.mathsisfun.com/alg
ebra/quadratic­equation­
derivation.html
For ax2 + bx + c = 0, x = -b ± √b2 - 4ac
2a
Solve by using the quadratic formula.
1.) t2 - 3t - 7 = 0
You can check problems
with real solutions using
your graphing calculator
2.) 4x2 + 8x + 31 = 0
Pre­Calculus
Section 4.2: Solving Quadratics Using the Quadratic Formula
February 17, 2016
Complex Numbers
Imaginary i, 5 + 2i
Real
­12, 5, 6/7, √3, ∛12
Rational ­ can be made into a fraction
­12, 5, 6/7
Irrational ­ cannot be made into a fraction
√3, ∛12, 1 + √2
7
Pre­Calculus
Section 4.2: Solving Quadratics Using the Quadratic Formula
February 17, 2016
Discriminant = b2 - 4ac
*the number under the radical in the quadratic formula.
Tells us...
• If the roots are real or imaginary
• If there are 0, 1, or 2 real roots.
• If the real roots are rational or irrational
All WITHOUT solving!
Discriminant
2
b - 4ac > 0
*positive
Nature of Roots/Zeros
2 real roots
• rational if it's a
perfect square
• irrational if it's not a
perfect square
b2 - 4ac = 0
b2 - 4ac < 0
*negative
Pre­Calculus
1 real root
• always rational
2 imaginary
roots
Graph
Section 4.2: Solving Quadratics Using the Quadratic Formula
February 17, 2016
Find the discriminant. Then state the nature of the
roots.
1.) x2 - 2x - 35 = 0
2.) x2 - 12x - 10 = 0
3.) 4x2 + 4x + 24 = 0
Pre­Calculus
Section 4.2: Solving Quadratics Using the Quadratic Formula
February 17, 2016
All imaginary roots come in pairs known as
conjugates known as a + bi and a - bi
Find the conjugate of each root
1. 6i
2. -1 + i
3. 7 - i√2
Pre­Calculus