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Quiz 9
Factor the following polynomials using integers. If it is not possible, state so.
note: all these can be solved with the quadratic formula; I chose not to use it when possible but it will always work!
1.) Solve x2+1=0.
Solution: Subtract 1 to get
x2=−1.
Now take square roots of each side to get
x=±√−1=±i.
2.) Solve x2+2x+1=0.
Solution: The left-hand side factors to give us
(x+1)(x+1)=0.
This yields two equations (which happen to be the same): x+1=0 and x+1=0. Hence the only solution is x=−1 (which appears twice).
3.) Solve x2−16=0.
Solution: Add 16 to both sides to get
x2=16.
Now take the square root of each side to get
x=±√16=±4.
4.) Solve 5x2−3x+2=0.
Solution: For this we use the quadratic formula with a=5, b=−3, and c=2 to get
x=−(−3)±√(−3)2−4(5)(2)2⋅5=3±√9−4010=3±√−3110=3±√31i10=310±√3110i.
5.)Solve −3x2+x+4=0.
Solution: The left-hand side factors and we get
−(x+1)(3x−4)=0,
hence the solution is x=−1,43.