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Math Practice

Nonlinear Systems Practice Problems

Substitute the simpler equation into the curved one, solve the quadratic, and find a partner for each root. Several problems only ask how many solutions there are, which the discriminant or a quick sketch can answer.

  1. 1.

    How many solutions does and have?

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    gives : two points, and .

  2. 2.

    Solve and .

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    , so or , each with .

  3. 3.

    Solve and .

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    gives , so and .

  4. 4.

    Solve and .

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    factors as . Partners from : and .

  5. 5.

    Solve and .

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    gives ; then .

  6. 6.

    Solve and .

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    gives , so or , with or .

  7. 7.

    Solve and .

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    gives , so or , giving and .

  8. 8.

    How many solutions does and have?

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    gives , both with : two solutions.

  9. 9.

    How many solutions does and have?

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    is : a double root, so the line is tangent at .

  10. 10.

    Does the line meet the parabola ? Find the number of intersection points.

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    has discriminant , so there are no intersection points.

  11. 11.

    How many solutions does and have?

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    Adding gives , so , and then , so : four points .

  12. 12.

    A rectangle has area 48 m² and perimeter 28 m. What is its length (the longer side), in meters?

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    and give , so the sides are 6 and 8. The length is 8 m.
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MPhil in Mathematics, Founder of Mathovia
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Nonlinear Systems Practice Problems

These problems follow the nonlinear systems lesson. Expect more than one solution, and pair each -value with its own -value.

Counting with the discriminant

Intersection points
positive2
zero1 (tangent)
negative0

Common mistakes to watch for

  • Missing solutions. A line and a parabola can meet twice; solve the resulting quadratic fully.
  • Forgetting to find the other coordinate. Each solution is a pair: substitute every back to find its .
  • Not checking for extraneous solutions. Squaring or substituting can introduce answers that fail the original equations.
  • Assuming there is always a solution. A negative discriminant means the curves never meet.

Frequently Asked Questions

How many solutions can a nonlinear system have?+

A line and a parabola or circle meet in 0, 1 or 2 points. A circle and a hyperbola can meet in up to 4.

How do I count solutions without solving?+

Substitute to get a quadratic and check the discriminant: positive gives 2, zero gives 1, negative gives 0.

How do I type several points?+

Separate them with "and", for example (-2, 0) and (3, 5). Either order is accepted.

More practice problems

Substitution Method Practice Problems Systems of Equations Practice Problems Systems of Inequalities Practice Problems