Nonlinear equations may have multiple solutions, one repeated solution, or no real solution. The most efficient method depends on the equation’s structure: factoring, taking roots, substitution, or using equivalent forms.
Essential concepts
\[x= rac{-bpmsqrt{b^2-4ac}}{2a}\]
For ax² + bx + c = 0, the discriminant b² − 4ac indicates two, one, or no real solutions.
SAT strategy and application
Solve a factorable quadratic
Solve \(x^2-5x+6=0\).
- Factor
\((x-2)(x-3)=0\).
- Use the zero-product property
\(x-2=0\) or \(x-3=0\).
Solutions: \(x=2\) and \(x=3\).
Mistakes and traps
Check your understanding
What are the solutions to \(x^2=49\)?
- \(7\)
- \(-7\)
- \(pm7\)
- \(49\)
Show answer and explanation
Answer: \(pm7\)
Both \(7^2\) and \((-7)^2\) equal 49.
Key takeaways
What to remember
- Choose a solution method based on visible equation structure.
- Set quadratics equal to zero before using the zero-product property.
- Account for every valid solution.
- Check for extraneous or excluded values in the original equation.
Continue learning
Put these notes into practice
Apply the ideas with SAT-style questions, then reinforce key details with flashcards.