Quadratics 40 points A quadratic contains a squared variable (x2) as the highest-order term
Standard form: 2 y=ax +bx+c There can be 0, 1, or 2 solutions for x (also called roots)
Thats because solutions are where the graph of an 0 solutions 2 solutions
1 solution To solve: 1st: Always move everything to one side of the equation and set it equal to zero! 2nd: Use one of the following techniques...
OPTION 1: Factoring aka: reverse-FOIL This is ALWAYS easiest if a=1, so try to simplify your expression so that this is the case!
HINT: nice-looking numbers in answer choices= clue that factoring is possible For x2+bx+c: Look for the factors of c that add up to equal b For ax2+bx+c : multiply a times c, look for the factors of ac that add up to b, use those factors to break the middle term into two pieces, then factor by grouping Check your work by FOILing!
OPTION 2: Square Rooting: Use when both sides of equation are perfect squares OPTION 2: Square Rooting example:
Ex: (x+3)2 = 49 You can use the square root on both sides because both sides of the equation are perfect squares! (x+3) = 7
x= 4 OPTION 3: Completing the Square Use with more difficult quadratic equations 1st: put the quadratic in standard form
2nd: divide b by 2 and square the result 3rd: add the result to both sides of the equation, factor, & solve by square rooting Another example...
OPTION 4: The Quadratic Formula *Complicated math, so only use as a last resort! The Quadratic Formula Use to find an exact solution
HINT: Use when you see radicals in the answer choices! *Be sure to write in standard form before plugging in the values of a, b, and c into the formula! MEMORIZE this formula!!!! Discriminant: the sign dictates the # of REAL solutions of the equation
+ = 2 real solutions 0 = 1 real solution - = 0 real solutions You MUST solve the discriminant 1st!
Graphing Parabolas Note: Quadratics (& functions) graph as parabolas (U-shaped) When a<0: will open DOWN When a>0: will open UP *Be sure you are familiar with the terms on the image
ANOTHER way to see them: Vertex Form y = a(x - h) 2
+k h & k are the x- and - coordinates of the parabolas vertex, respectively, and the equation x = h give the axis of symmetry From standard form, plug in the values into x = -b 2a
This is the quadratic formula WITHOUT the square root part! To find the y-coordinate of the vertex, plug the x-coordinate into the original equation and solve for y!