2x^(2) - 3x - 11 = 0
This does not factor , so apply the Quadratic Eq'n.
x = { -b +/- sqrt[b^(2) - 4ac]} / 2a
x = { - -3 +/- sqrt[(-3)^2 - 4(2)(-11)]} / 2(2)
x = { (+)3 +/-sqrt[9+88]} / 4
x = { 3 +/- sqrt[97]} / 4
x = { 3 +/-9.848857802 } / 4
x = 12.848857802/4 = 3,21221445
&
x = -6.848857802/4 = - 1.71221445
Hence the two approximate answers are 3.21 & -1.71
33
x = -1.5 or -2 2x2 + 7x + 6 = 0 => (2x + 3)(x + 2) = 0 Thus 2x + 3 = 0 => x = -1.5 or x + 2 = 0 => x = -2
use algebra and then factor... x= -9 and 9
2x2 - 8x + 3 = 0 needs to be solved using the quadratic formula.x = (4 + sqrt(10))/2 or x = (4 - sqrt(10))/2
f(x) = 2x2 + 4x - 7
It has roots x = 2.618 and x = 0.38197
(3,11)
(3,11) is a factor pair of 33.
33
2x2 - 9x = 18 2x2 - 9x -18 = 0 (2x - 18)(x + 9) x= -9 and 9
x = -1.5 or -2 2x2 + 7x + 6 = 0 => (2x + 3)(x + 2) = 0 Thus 2x + 3 = 0 => x = -1.5 or x + 2 = 0 => x = -2
2x2 + 2y - 8 = 0 2(x2 + y - 4) = 0 and other than removing the two outside the bracket, this cannot be factorised further, nor simplified.
3x11 = 33
Write down all the pairs of factors of '2' & '12' Hence 2 = ( 2,1) 12 = ( 1,12) , ( 2, 6), ( 3 ,4) From these pairs we select two sets of factors , one each from '1' & '12' , which multiply and then add to '5' They are ( 2,1) & ) 3.4) Multiply ( 4 x 2) & ( 3 x 1) = 8 & 3 ; 8 - 3 = 5 Opening brackets ( 2x 3)( x 4) ( 2x + 3)( x - 4) Or 2x^(2) - 5x = 12 Algebraically rearrange 2x^(2) - 5x - 12 = 0 It is now in Quadratic Form and can be factored. Which signs
use algebra and then factor... x= -9 and 9
2x2 - 8x + 3 = 0 needs to be solved using the quadratic formula.x = (4 + sqrt(10))/2 or x = (4 - sqrt(10))/2
The curve turns at a minimum: (2.5, -12)