Nonlinear Equations in 1 Variable and Systems of Equations in 2 Variables
5 min readMedium10-question skill check
Quadratics show up on nearly every Math section, and the test loves to ask you to solve them three or four different ways. Master a handful of techniques and these become some of the fastest points on the test.
Short walkthroughs on Nonlinear Equations in 1 Variable and Systems of Equations in 2 Variables — ours, plus curated picks. Watch one if a written explanation isn't clicking, or after to reinforce.
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Description of the figure:
The x-axis runs from negative 1 to 7 and the y-axis from negative 3 to 12, with grid lines every 1 unit.
The parabola opens upward; its vertex, the lowest point, is at (3.5, negative 2.25).
It crosses the y-axis at (0, 10) and meets the x-axis at (2, 0) and (5, 0).
It also passes through (1, 4) and (7, 10).
y = x² - 7x + 10 crosses the x-axis at x = 2 and x = 5 — those crossings are the solutions.
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Description of the figure:
The x-axis runs from negative 4 to 4 and the y-axis from negative 5 to 8, with grid lines every 1 unit.
The parabola opens upward; its vertex, the lowest point, is at (0, negative 3).
It crosses the y-axis at (0, negative 3) and meets the x-axis at about (negative 1.73, 0) and about (1.73, 0).
It also passes through (negative 3, 6) and (3, 6).
The line rises from left to right.
It passes through the origin at (0, 0).
It also passes through (negative 2, negative 4) and (4, 8).
The parabola and the line intersect at (negative 1, negative 2) and (3, 6).
The line y = 2x crosses the parabola y = x² − 3 at (−1, −2) and (3, 6): those points are the solutions of the system.
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Quick check
Check your understanding with a question from this topic:
What is the positive solution to the equation x² - 7x + 10 = 0?
Enter a whole number, fraction (e.g. 3/4), or decimal (e.g. .75).
Worked examples
Example 1
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Example 2
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Example 3
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Common pitfalls
Forgetting the ± when square-rooting
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Sign errors in factoring
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Not setting the equation to zero first
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Misreading 'the' solution
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Key takeaways
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Tracks your progress across lessons.
Skill check
10 questions on Nonlinear Equations in 1 Variable and Systems of Equations in 2 Variables, from our question bank. Score 70% or better across 10 answers to reach Silver and pass this step.
A nonlinear equation is any equation where the variable is raised to a power other than 1 (or sits under a square root, etc.). The most common type on the test is the quadratic equation — one with an x² term. The standard form is:
ax² + bx + c = 0
where a, b, and c are numbers and a ≠ 0. The solutions (also called roots or zeros) are the x-values that make the equation true. A quadratic can have two, one, or zero real solutions.
You have four main tools:
1. Factoring. Rewrite ax² + bx + c as a product of two binomials. For x² - 7x + 10, find two numbers that multiply to +10 and add to -7: those are -2 and -5. So it factors to (x-2)(x-5) = 0. By the zero-product property, if two things multiply to 0, at least one must be 0 — so x = 2 or x = 5.
2. The quadratic formula. Always works, even when factoring is ugly:
x = (-b ± √(b² - 4ac)) / (2a)
The piece under the root, b² - 4ac, is the discriminant. If it's positive → 2 solutions; zero → 1 solution; negative → no real solutions.
3. Vieta's shortcuts. For ax² + bx + c = 0, the sum of the roots is -b/a and the product is c/a. If a question asks only for the sum or product, you don't need to solve — just read off the coefficients.
4. Square-rooting. If there's no x term, like x² = 49, take the square root of both sides — but remember both signs: x = ±7.
Systems with a nonlinear equation: sometimes you get a line and a parabola together, like y = x² - 3 and y = 2x. Substitution is your friend: set them equal (x² - 3 = 2x), move everything to one side (x² - 2x - 3 = 0), and solve the quadratic. Each solution is the x of a point where the graphs cross.
Graphically, the solutions are exactly the points where the curves intersect.
What are the solutions to x² - 12x + 32 = 0?
If x² - 8x + 7 = 0, what is the sum of the two solutions?
The system of equations below has how many real solutions (x, y)?
y = x² + 2
y = 6x - 7
From x² = 49, students write only x = 7 and miss x = -7. Every time you take a square root to solve, both the positive and negative roots are solutions.
For x² - 7x + 10, the factors are (x-2)(x-5), so the roots are +2 and +5, not -2 and -5. The root is the value that makes the bracket zero — flip the sign of the number inside.
Factoring and the quadratic formula only work when one side is 0. If you see x² + 2 = 6x - 7, you must move everything to one side before factoring or applying the formula.
A quadratic usually has two solutions. If a question asks for the positive solution or the larger value, identify which one it wants — don't just bubble the first root you find.
Standard form is ax² + bx + c = 0; the solutions are the x-values that make it true.
Try factoring first; fall back to the quadratic formula x = (-b ± √(b²-4ac))/(2a) when factoring is hard.
The discriminant b² - 4ac tells you the number of real solutions: positive → 2, zero → 1, negative → 0.
Vieta's: sum of roots = -b/a, product of roots = c/a — use these to skip solving.
For a line-and-curve system, substitute to get one quadratic; intersection points = real solutions.