Hey guys, Adrian here
First thing in class, Mr.P explained us how to solve some of the questions from the worksheet that he gave us last class. One of the question is asin²x - 56 = -25sinx which has no solution. After that, we learned how to derive the Quadratic Formula by completing the square.
This is how we did it:
ax² + bx +c = 0 is the standard form.
steps:
Divide it by a. ax² + bx + c = 0
a
x² + bx + c
a a
Isolate terms with x x² + bx = _ c
on one side. a a
Complete the square by adding (b/a / 2)² = (b/a * 1/2)² = (b/2a)²
((1/2)(b/a))²
x² + bx/a + (b/2a)² = -c/a + (b/2a)²
x² + bx + b² = _ c + b²
a 4a² a 4a²
Simplify. Factor the left side (x + b/2a)² = -c/a + b²/4a²
and square the term on the right.
(b/2a)² = b²/4a²
Simplify. (Common denominator for the (x + b/2a)² = -4ac/4a² + b²/ 4a²
terms on the right.)
Take the square root of each side. √(x + b/2a)² = √b²-4ac/4a²
Subtract (b/2a) from each side . x + b/2a = √b² - 4ac/ 4a²
( notice in the denominator the square
root of 4a² is 2a). x = -b/2a + or - √b² - 4ac /2a
So the Quadratic formula will be:
x = -b/2a + or - √b² - 4ac/ 2a
Example using the quadratic formula:
1.) 2x² - 6x - 7 = 0
a = 2 , b = -6 , c = -7
x = -b + √b² - 4ac x = -b - √b² - 4ac
2a 2a
= -(-6) + √(-6)² - 4*2(-7) = -(-6) - √(-6)² - 4*2(-7)
2(2) 2(2)
= 6 + √36 + 56 = 6 - √36 + 56
4 4
= 6 + √92 = 6 - √92
4 4
x = 3.8979 x = - 0.897
-> okay thats it.. =) dont forget to work on the worksheet he gave us..
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