The set of solutions for the equation $\left(x^{2}-25\right)^{2}=x^{2}-10 x+25$ contains how many real numbers?

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The set of solutions for the equation (x^{2} - 25)^{2} = x^{2} - 10 x + 25 contains how many real numbers?

Explanation	Calculations	Help
$We notice that the right-hand side is the square of x minus 5.$	$(x^{2} - 25)^{2} = (x - 5)^{2}$	Theory & Tactics Method Card ALG4-A Click to view full details
$We move all terms to one side to create a difference of squares.$	$(x^{2} - 25)^{2} - (x - 5)^{2} = 0 ((x^{2} - 25) - (x - 5)) \times ((x^{2} - 25) + (x - 5)) = 0$	Theory & Tactics Method Card ALG4-B Click to view full details
$We set the first factor equal to zero and solve for x.$	$((x^{2} - 25) - (x - 5)) = 0 x^{2} - x - 20 = 0 (x - 5) (x + 4) = 0 x = 5, - 4$	Theory & Tactics Method Card EQUA2-B Click to view full details
$We set the second factor equal to zero and solve for x.$	$((x^{2} - 25) + (x - 5)) = 0 x^{2} + x - 30 = 0 (x + 6) (x - 5) = 0 x = - 6, 5$	Theory & Tactics Method Card EQUA2-B Click to view full details
$We collect the distinct real solutions.$	$x = - 6, - 4, 5$

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The set of solutions for the equation (x^{2} - 25)^{2} = x^{2} - 10 x + 25 contains how many real numbers?

Explanation	Calculations	Help
$We notice that the right-hand side is the square of x minus 5.$	$(x^{2} - 25)^{2} = (x - 5)^{2}$	Theory & Tactics Method Card ALG4-A Click to view full details
$We move all terms to one side to create a difference of squares.$	$(x^{2} - 25)^{2} - (x - 5)^{2} = 0 ((x^{2} - 25) - (x - 5)) \times ((x^{2} - 25) + (x - 5)) = 0$	Theory & Tactics Method Card ALG4-B Click to view full details
$We set the first factor equal to zero and solve for x.$	$((x^{2} - 25) - (x - 5)) = 0 x^{2} - x - 20 = 0 (x - 5) (x + 4) = 0 x = 5, - 4$	Theory & Tactics Method Card EQUA2-B Click to view full details
$We set the second factor equal to zero and solve for x.$	$((x^{2} - 25) + (x - 5)) = 0 x^{2} + x - 30 = 0 (x + 6) (x - 5) = 0 x = - 6, 5$	Theory & Tactics Method Card EQUA2-B Click to view full details
$We collect the distinct real solutions.$	$x = - 6, - 4, 5$

Scroll horizontally to view all columns

3