Math/Elective Algebra
Quadratic Equations
ax2 + bx + c = 0
Examples of Quadratic equations
1. x2 +2x – 8 = 0 2. x2 – 10x + 25 = 0 3. 3x2 + x - 2 = 0
Quadratic Formula
If [pic] a x2 + b x + c = 0, then [pic]
Finding the zeros of the quadratic functions
- The zeros of a function are the input values which result in an output value of zero.
One way of solving quadratic equations is using factoring
Examples are the following:
1) x2 + 5x + 6 = 0 Set this equal to zero: (x + 2)(x + 3) = 0 Solve each factor: -2008 All Rights Reserved x + 2 = 0 or x + 3 = 0 x = –2 or x = – 3
Then, clearly, the zeros are -2 and -3, since those are the input values which will result in an output value of zero.
2) x(x + 5) = 0 3) x2 – 4 = 0 x = 0 or x + 5 = 0 (x – 2) (x + 2) = 0 x = 0 or x = –5 x – 2 = 0 or x + 2 = 0 x = 2 or x = –2
Another way of solving quadratic equations is using quadratic formula
Given, f ( x ) = 2 x2 + 8 x + 1, the zeros are the values of x which result in an output value of y = 0. Thus the zeros are the solutions of the equation 2 x2 + 8 x + 1 = 0 Using the quadratic formula, we get
[pic]
Another way of solving quadratic equations is using completing the square
x2 + 6x – 7 = 0 x2 + 6x = 7 x2 + 6x + 9 = 7 + 9
(x+3)2 = 16 x + 4 = ± 4 x = – 3 ± 4 = – 3 – 4, –3 + 4 = –7,
ax2 + bx + c = 0
Examples of Quadratic equations
1. x2 +2x – 8 = 0 2. x2 – 10x + 25 = 0 3. 3x2 + x - 2 = 0
Quadratic Formula
If [pic] a x2 + b x + c = 0, then [pic]
Finding the zeros of the quadratic functions
- The zeros of a function are the input values which result in an output value of zero.
One way of solving quadratic equations is using factoring
Examples are the following:
1) x2 + 5x + 6 = 0 Set this equal to zero: (x + 2)(x + 3) = 0 Solve each factor: -2008 All Rights Reserved x + 2 = 0 or x + 3 = 0 x = –2 or x = – 3
Then, clearly, the zeros are -2 and -3, since those are the input values which will result in an output value of zero.
2) x(x + 5) = 0 3) x2 – 4 = 0 x = 0 or x + 5 = 0 (x – 2) (x + 2) = 0 x = 0 or x = –5 x – 2 = 0 or x + 2 = 0 x = 2 or x = –2
Another way of solving quadratic equations is using quadratic formula
Given, f ( x ) = 2 x2 + 8 x + 1, the zeros are the values of x which result in an output value of y = 0. Thus the zeros are the solutions of the equation 2 x2 + 8 x + 1 = 0 Using the quadratic formula, we get
[pic]
Another way of solving quadratic equations is using completing the square
x2 + 6x – 7 = 0 x2 + 6x = 7 x2 + 6x + 9 = 7 + 9
(x+3)2 = 16 x + 4 = ± 4 x = – 3 ± 4 = – 3 – 4, –3 + 4 = –7,