Step-by-step explanation:
The discriminant of the quadratic equation [tex]ax^2+bx+c=0[/tex]:
[tex]\Delta=b^2-4ac[/tex]
If Δ < 0, then the equation has two complex roots [tex]x=\dfrac{-b\pm\sqrt\Delta}{2a}[/tex]
If Δ = 0, then the equation has one repeated root [tex]x=\dfrac{-b}{2a}[/tex
If Δ > 0, then the equation has two discint roots [tex]x=\dfrac{-b\pm\sqrt\Delta}{2a}[/tex]
[tex]1.\ x^2-4x+2=0\\\\a=1,\ b=-4,\ c=2\\\\\Delta=(-4)^2-4(1)(2)=16-8=8>0,\ \bold{two\ distinct\ roots}\\\sqrt\Delta=\sqrt8=\sqrt{4\cdot2}=2\sqrt2\\\\x=\dfrac{-(-4)\pm2\sqrt2}{2(1)}=\dfrac{4\pm2\sqrt2}{2}=2\pm\sqrt2\\\\==============================\\\\2.\ 5x^2-2x+3=0\\\\a=5,\ b=-2,\ c=3\\\\\Delta=(-2)^2-4(5)(3)=4-60=-56<0,\ \bold{two\ complex\ roots}\\\sqrt\Delta=\sqrt{-56}=\sqrt{(-4)(14)}=2\sqrt{14}\ i\\\\x=\dfrac{-(-2)\pm2\sqrt{14}\ i}{(2)(5)}=\dfrac{1\pm\sqrt{14}\ i}{5}\\\\==============================[/tex]
[tex]3.\ 2x^2+x-6=0\\\\a=2,\ b=1,\ c=-6\\\\\Delta=1^2-4(2)(-6)=1+48=49>0,\ \bold{two\ distinct\ roots}\\\sqrt\Delta=\sqrt{49}=7\\\\x=\dfrac{-1\pm7}{(2)(2)}\\\\x_1=\dfrac{-8}{4}=-2,\ x_2=\dfrac{6}{4}=\dfrac{3}{2}\\\\==============================\\\\4.\ 13x^2-4=0\qquad\text{add 4 to both sides}\\\\13x^2=4\qquad\text{divide both sides by 13}\\\\x^2=\dfrac{4}{13}\to x=\pm\sqrt{\dfrac{4}{13}},\ \bold{two\ distinct\ roots}\\\\==============================[/tex]
[tex]5.\ x^2-6x+16=0\\\\a=1,\ b=-6,\ c=16\\\\\Delta=(-6)^2-4(1)(16)=36-64=-28<0,\ \bold{two\ complex\ roots}\\\sqrt\Delta=\sqrt{-28}=\sqrt{(-4)(7)}=2\sqrt7\ i\\\\x=\dfrac{-(-6)\pm2\sqrt7\ i}{(2)(1)}=3\pm\sqrt7\ i\\\\==============================\\\\6.\ x^2-8x+16=0\\\\a=1,\ b=-8,\ c=16\\\\\Delta=(-8)^2-4(1)(16)=64-64=0,\ \bold{one\ repea}\bold{ted\ root}\\\\x=\dfrac{-(-8)}{(2)(1)}=\dfrac{8}{2}=4\\\\==============================\\\\[/tex]
[tex]7.\ 4x^2+11=0\qquad\text{subtract 11 from both sides}\\\\4x^2=-11\qquad\text{divide both sides by 4}\\\\x^2=-\dfrac{11}{4}\to x=\pm\sqrt{-\dfrac{11}{4}}\\\\x=\pm\dfrac{\sqrt{11}}{2}\ i,\ \bold{two\ complex\ roots}[/tex]
Answer:
For the equation x2 − 4x + 2 = 0, the discriminant is (-4)2 − 4(1)(2) = 8. Since the discriminant is positive, it has two distinct real roots.
For the equation 5x2 − 2x + 3 = 0, the discriminant is (-2)2 − 4(5)(3) = -56. Since the discriminant is negative, it has two complex roots.
For the equation 2x2 + x − 6 = 0, the discriminant is (1)2 − 4(2)(-6) = 49. Since the discriminant is positive, it has two distinct real roots.
For the equation 13x2 − 4 = 0, the discriminant is (0)2 − 4(13)(-4) = 208. Since the discriminant is positive, it has two distinct real roots.
For the equation x2 − 6x + 9 = 0, the discriminant is (-6)2 − 4(1)(9) = 0. Since the discriminant is zero, it has one repeated root.
For the equation x2 − 8x + 16 = 0, the discriminant is (-8)2 − 4(1)(16) = 0. Since the discriminant is zero, it has one repeated root.
For the equation 4x2 + 11 = 0, the discriminant is (0)2 − 4(4)(11) = -176. Since the discriminant is negative, it has two complex roots.
