Using the Quadratic Formula to Solve a Quadratic Equation

But this parabola has two zeros. Example Use the Zero Product Property to find the roots of: Product But this parabola has two zeros. (x2)(-7) -7x2 c -7 ax2c IMPOSSIBLE x2 bx ___ ax2 -3x Sum Just because a quadratic is not factorable, does not mean it does not have roots. Thus, there is a need for a new algebraic method to find these roots.

Quadratic Formula For ANY 0 = ax2 + bx +c (standard form) the value(s) of x is given by: MUST equal 0 Plus or Minus Opposite of b “All Over”

Solving a Quadratic with the Quadratic Formula Algebraically solve: Must equal 0 Find the values of “a,” “b,” “c” a = b = c = 1 -3 -7 Simplify the expression in the square root first Since the square root can not simplified, this is an acceptable EXACT answer Substitute into the Quadratic Formula Or you can approximate the expressions (don’t forget parentheses). This is NOT exact. Or you can write two expressions. One with addition in the numerator and other with subtraction. This is also an EXACT answer.

Solving a Quadratic with the Quadratic Formula Algebraically solve: Must equal 0 Find the values of “a,” “b,” “c” a = b = c = 2 6 -5 Simplify the expression in the square root first The square root can be simplified. Substitute into the Quadratic Formula The GCF of every term is 2

Solving a Quadratic: Make Sure to Isolate 0 Solve: Find the values of “a,” “b,” “c” Solve for 0 first! a = b = c = 1 -3 -4 Distribute Simplify the expression in the square root first The square root can be simplified. Substitute into the Quadratic Formula Or Since the answers will be rational, it is best to list both.

Solving a Quadratic with the Quadratic Formula: Two Solutions Algebraically solve: Must equal 0 Find the values of “a,” “b,” “c” a = b = c = 4 -121 Simplify the expression in the square root first The square root can be simplified. Substitute into the Quadratic Formula Or Since the answers will be rational, it is best to list both.

Solving a Quadratic with the Quadratic Formula: No Solutions Algebraically solve: Must equal 0 Find the values of “a,” “b,” “c” a = b = c = 1 -5 9 Simplify the expression in the square root first Substitute into the Quadratic Formula This can not be calculated because you can not square root a negative. The graph of the quadratic has no x-intercepts. NO SOLUTIONS

Solving a Quadratic with the Quadratic Formula: One Solution Algebraically solve: Must equal 0 Find the values of “a,” “b,” “c” a = b = c = 36 -60 25 Simplify the expression in the square root first The square root of 0 is 0. Substitute into the Quadratic Formula The is no difference to adding zero or subtracting 0. This expression will result in only one answer.

The Discriminant of a Quadratic For ANY 0 = ax2 + bx +c (standard form) the value given by: If the Discriminant (the value underneath the square root in the quadratic formula) is…. Greater than zero there are two roots Equal to zero there is one root Less than zero there are no roots