The cube root is already alone, so cube both sides. A cube root undoes a cube, and every real number has exactly one cube root, so this step keeps exactly the same solutions.
x3+7=(x+1)3
2
Expand the cube
(x+1)3=x3+3x2+3x+1. Write all four terms, since the middle two are the ones most often lost.
x3+7=x3+3x2+3x+1
3
Collect on one side
Subtract x3+7 from both sides. The x3 terms cancel, which leaves a quadratic.
3x2+3x−6=0
4
Divide by 3 and factor
Dividing every term by 3 gives x2+x−2=0, which factors as (x+2)(x−1)=0.
x2+x−2=0(x+2)(x−1)=0
5
Write both solutions
Each factor gives one solution, x=−2 or x=1. Cubing could not have added a false one.
x=−2 or x=1
6
Check both anyway
At x=1,3√1+7=3√8=2 and 1+1=2. At x=−2,3√−8+7=3√−1=−1and −2+1=−1. A cube root of a negative number is a real negative number, so x=−2 is fine.
3√−8+7=3√−1=−1
Result
x=−2 or x=1. Both check, as they must after cubing.
x=−2 or x=1
Your turn
1. One like the example
Solve 3√5x+2=−3.
Show the answer and explanation
x=−29/5.
Cube both sides to get 5x+2=−27. Then 5x=−29 and x=−29/5, which is −5.8. Check: 5(−5.8)+2=−27, and 3√−27=−3.
3√5x+2=−35x+2=−275x=−29x=−529
2. A variation
Solve x3/2=8.
Show the answer and explanation
x=4.
x3/2 means (√x)3. The cube of √x is 8, so √x=2, and squaring gives x=4. Check: 43/2=(√4)3=23=8.
x3/2=8(√x)3=8√x=2x=4
3. Somewhere new
For a planet orbiting the Sun, T2=a3, with T its year in Earth years and a its distance in astronomical units. A planet’s year lasts 8 Earth years. How far is it from the Sun?
Show the answer and explanation
a=4 astronomical units.
Put T=8 into T2=a3 to get a3=64. The real cube root of 64 is 4, and cubing has only that one real solution. Check: 43=64=82.
a3=82a3=64a=4
Keep exploring
Open the steps in Math and remove x=−2 from the last row. The Checker marks it ✗, because cubing kept both solutions.
