Solving Quadratic Equations

Solving Quadratic Equations A quadratic equation is one which can be written in the form ax 2 + bx + c = 0 where a ≠ 0 . A solution to any equation is a value which makes the equation true. A quadratic equation is also called a second degree polynomial equation. The degree of a polynomial equation is an upper bound for the number of solutions to the equation. Hence, a quadratic equation can have no more than two solutions. Types of quadraticsSolving by factoring The key idea behind solving by factoring is that if a product is equal to zero, then one or both of the factors must equal zero. It is important to remember that this only works if one side of the equation is zero. ( x − a )( x − b) = 0 ⇒ x − a = 0, x − b = 0 ⇒ x = a, b.

Examples: x 2 − 9 = 0 ⇒ ( x + 3)( x − 3) = 0 ⇒ x + 3 = 0, x − 3 = 0 ⇒ x = ±3. x 2 − 4 x − 5 = 0 ⇒ ( x − 5)( x + 1) = 0 ⇒ x − 5 = 0, x + 1 = 0 ⇒ x = 5,−1. 1 3x 2 + 5 x − 2 = 0 ⇒ (3 x − 1)( x + 2) = 0 ⇒ 3 x − 1 = 0, x + 2 = 0 ⇒ x = ,−2. 3

If one side of the equation is not zero, then add or subtract as need to get a zero on one side. x 2 + x = 20 ⇒ x 2 + x − 20 = 0 ⇒ ( x + 5)( x − 4) = 0 ⇒ x = −5,4.

The following does not work because on side of the equation is not zero. x 2 + 2 x = 5 ⇒ x( x + 2) = 5 ⇒ x = 5, x + 2 = 5 ⇒ x = 5,3 and neither of these values satisfy the original equation. A quadratic can always be factored, but at times it is worth neither the time nor the effort since there are other techniques that always work and factoring can often be difficult.

x2 = k .

This type of equation can be solved by taking both positive and negative square roots of the right hand side. Ex.: x 2 = 9 ⇒ x = ± 9 = ±3 . The symbol “ ⇒ ” stands for implies. ( x − h) 2 = k ⇒ x − h = ± k ⇒ x = h ± k

.

Ex.: ( x + 3) = 4 ⇒ x + 3 = ±2 ⇒ x = −3 ± 2 ⇒ x = −3 + 2,−3 − 2 ⇒ x = −1,−5 . Before taking the square roots of the right hand side, it can be desirable to isolate the square by dividing both sides by whatever precedes it. 2

a ( x − h) 2 = k ⇒ ( x − h ) 2 =

k k k ⇒ x−h = ± ⇒x =h± . a a a

9( x + 2) 2 = 16 ⇒ ( x + 2) 2 =

Ex.:

− 2 − 10 , . 3 3

16 16 4 −6 +4 −6 −4 ⇒ x+2 = ± ⇒ x = −2 ± = , = 9 9 3 3 3