Nonlinear Equations in 1 Variable and Systems of Equations in 2 Variables
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.
Watch & learn
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.
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.
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.
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).
- Marked points: (negative 1, negative 2), labeled (−1, −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.
Graphically, the solutions are exactly the points where the curves intersect.
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
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
Common pitfalls
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.
Key takeaways
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 - 4actells 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.
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.