MATH SOLVE

4 months ago

Q:
# Which describes the number and type of roots of the equation x^4 - 64 = 0a. 2 real roots, 2 imaginary rootsb. 4 real rootsc. 3 real roots, 1 imaginary rootd. 4 imaginary roots

Accepted Solution

A:

ANSWERa. 2 real roots, 2 imaginary rootsEXPLANATIONThe given equation is [tex] {x}^{4} - 64 = 0[/tex]We rewrite as difference of two squares,[tex]( {x}^{2} )^{2} - {8}^{2} = 0[/tex]We factor using difference of two squares to get;[tex]( {x}^{2} - 8)( {x}^{2} + 8) = 0[/tex]We now use the zero product property to get:[tex]{x}^{2} = 8 \: or \: {x}^{2} = - 8[/tex]Take the square root of both sides to get;[tex]{x} = \pm \sqrt{8} \: or \: {x}^{2} = \pm \sqrt{ - 8} [/tex][tex]{x} = \pm 2\sqrt{2} \: or \: {x} = \pm 2\sqrt{ 2} i[/tex][tex]{x} = - 2\sqrt{2} \: or \: {x} = 2\sqrt{ 2}[/tex]are two real roots.[tex]{x} = - 2\sqrt{2}i \: or \: {x} = 2\sqrt{ 2} i[/tex]are two imaginary roots.The correct answer is A.