Without solving each equation, determine the number of solutions. The equations are: (a)4x2-12x+9=0 (b)2x2+5x-1=0 (c)x2-2x+3=0
Use the determinant b(squared) - 4ac
if it is>0 then 2 real solutions
if it =0 then 1 real solution
if it<0 2 imaginary solutions
a)(-12)(squared) - 4(4)(9) = 0 so: 1 real
b)5(squared) - 4(2)(-1) = 33 so: 2 real
c)(-2)(squared) - 4(1)(3) =-8 so: 2 imaginary
A quadratic equation of the type ax^2+bx+c=0 has the discriminant, D = b^2-4ac. The nature roots of this general quadratic equation dependant upon the discriminant D.The roots are (i)two distinct real roots if D >0.(ii) The two roots are identical and merging in one single root if D = 0 and (iii) the two roots are conjugate imaginary , if D <0.
Therefore, applying this rule to (a), B) and (c), we get:
a) The discriminant D = (-12)^2-4*4*9 = 144-144. So ,the roots are identical.
b)The discriminant, D = 5^2-2*2*(-1) = 29 +ve. The roots are two different real roots that are also a pair of irrational roots, as square root of 29 is irrationals or surds.
c)The discrimnant, D = (-2)^2-4*1*3 = 4-12 =-8. S o the roots are a pair imaginary nynmbers,(or also apair of conjugate complex numbers).