# Problem of the Week

## Updated at May 22, 2023 9:05 AM

This week's problem comes from the equation category.

How would you solve $${({(u-3)}^{2}+2)}^{2}=9$$?

Let's begin!

${({(u-3)}^{2}+2)}^{2}=9$

 1 Take the square root of both sides.${(u-3)}^{2}+2=\pm \sqrt{9}$2 Since $$3\times 3=9$$, the square root of $$9$$ is $$3$$.${(u-3)}^{2}+2=\pm 3$3 Break down the problem into these 2 equations.${(u-3)}^{2}+2=3$${(u-3)}^{2}+2=-3$4 Solve the 1st equation: $${(u-3)}^{2}+2=3$$.1 Subtract $$2$$ from both sides.${(u-3)}^{2}=3-2$2 Simplify  $$3-2$$  to  $$1$$.${(u-3)}^{2}=1$3 Take the square root of both sides.$u-3=\pm \sqrt{1}$4 Simplify  $$\sqrt{1}$$  to  $$1$$.$u-3=\pm 1$5 Break down the problem into these 2 equations.$u-3=1$$u-3=-1$6 Solve the 1st equation: $$u-3=1$$.1 Add $$3$$ to both sides.$u=1+3$2 Simplify  $$1+3$$  to  $$4$$.$u=4$To get access to all 'How?' and 'Why?' steps, join Cymath Plus!$u=4$7 Solve the 2nd equation: $$u-3=-1$$.1 Add $$3$$ to both sides.$u=-1+3$2 Simplify  $$-1+3$$  to  $$2$$.$u=2$To get access to all 'How?' and 'Why?' steps, join Cymath Plus!$u=2$8 Collect all solutions.$u=4,2$To get access to all 'How?' and 'Why?' steps, join Cymath Plus!$u=4,2$5 Solve the 2nd equation: $${(u-3)}^{2}+2=-3$$.1 Subtract $$2$$ from both sides.${(u-3)}^{2}=-3-2$2 Simplify  $$-3-2$$  to  $$-5$$.${(u-3)}^{2}=-5$3 Take the square root of both sides.$u-3=\pm \sqrt{-5}$4 Simplify  $$\sqrt{-5}$$  to  $$\sqrt{5}\imath$$.$u-3=\pm \sqrt{5}\imath$5 Break down the problem into these 2 equations.$u-3=\sqrt{5}\imath$$u-3=-\sqrt{5}\imath$6 Solve the 1st equation: $$u-3=\sqrt{5}\imath$$.1 Add $$3$$ to both sides.$u=\sqrt{5}\imath +3$2 Regroup terms.$u=3+\sqrt{5}\imath$To get access to all 'How?' and 'Why?' steps, join Cymath Plus!$u=3+\sqrt{5}\imath$7 Solve the 2nd equation: $$u-3=-\sqrt{5}\imath$$.1 Add $$3$$ to both sides.$u=-\sqrt{5}\imath +3$2 Regroup terms.$u=3-\sqrt{5}\imath$To get access to all 'How?' and 'Why?' steps, join Cymath Plus!$u=3-\sqrt{5}\imath$8 Collect all solutions.$u=3+\sqrt{5}\imath ,3-\sqrt{5}\imath$To get access to all 'How?' and 'Why?' steps, join Cymath Plus!$u=3+\sqrt{5}\imath ,3-\sqrt{5}\imath$6 Collect all solutions.$u=4,2,3+\sqrt{5}\imath ,3-\sqrt{5}\imath$Doneu=4,2,3+sqrt(5)*IM,3-sqrt(5)*IM