We are going to use the quadratic equation ax^{2} + bx + c = 0
REMEMBER: b^{2 }- 4ac is the discriminant
Doesn't make sense? Here's an example.
Find the value of the discriminant and use that value to tell if the equation has two solutions, one solution, or no solutions.
a. x^{2 }- 3x - 4 = 0
1st STEP: Find a, b and c values.
a: 1 b: 2 c: -4
2nd STEP: Plug a, b and c into the discriminant
b^{2 }- 4ac = (-3)^{2 }- 4(1)(-4)
= 9 + 16
= 25
* It equals 25, which is a positive number so the equation has TWO solutions.
You and a friend are camping. You want to hang your food pack from a branch 20 feet from the ground. You will attach a rope to a stick and throw it over the branch.
a. Your friend can throw the stick upward with an initial velocity of 29 feet per second from an initial height of 6 feet. Will the stick reach the branch when it is thrown?
** Use the vertical motion model: h = -16t ^{2}+ vt + s
( h= height you are trying to reach, t= time in motion, v= initial velocity, and s= the initial height )
1ST STEP: Use the information from the word problem, and plug it into the appropriate places in the equation!
20 = -16t^{2} + 29t + 6
2ND STEP: Rewrite the equation in Standard Form. (ax^{2}^{ }+ bx + c) Remember in standard form we always want our equations to equal 0.
20 = -16t^{2} + 29t + 6
-20 = -16t^{2} + 29t + 6 - 20
0 = -16t^{2} + 29t - 4
3RD STEP: Find using the discriminant!
0 = -16t^{2} + 29t - 4
a: - 16 b: 29 c: - 4
29^{2} - 4(-16)(-4)
841 - 896 = - 55
** The answer is negative, which means there are no solutions, and no x-intercepts. The stick thrown by your friend did not make it over the tree branch.