Question 1038065
Let {{{ z = x^2 }}}
then {{{ x^4 = z^2 }}}
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Rewrite the equation:
{{{ z^2 - 52z + 576 = 0 }}}
Complete the square:
{{{ z^2 - 52z + ( 52/2 )^2 = ( 52/2 )^2 - 576 }}}
{{{ z^2 - 52z + 676 = 676 - 576 }}}
{{{ z^2 - 52z + 676 = 100 }}}
{{{ ( z - 26 )^2 = 10^2 }}}
Take the square root of both sides
{{{ z - 26 = 10 }}}
{{{ z = 36 }}}
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also, using the negative square root of {{{ 100 }}},.
{{{ z - 26 = -10 }}}
{{{ z = 16 }}}
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Now go back to
{{{ x^2 = z }}}
{{{ x^2 = 36 }}}
{{{ x = 6 }}}
{{{ x = -6 }}}
and
{{{ x^2 = 16 }}}
{{{ x = 4 }}}
{{{ x = -4 }}}
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I get 4 real solutions,
{{{ x = 4 }}}
{{{ x = -4 }}}
{{{ x = 6 }}}
{{{ x = -6 }}}
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You can check these by plugging back into equation
Here's the plot:
{{{ graph( 400, 400, -10, 10, -150, 700, x^4 - 52x^2 + 576 ) }}}