Math

Nonlinear Equations in 1 Variable and Systems of Equations in 2 Variables

2 min readMedium5-question drill

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.

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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.

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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.

Where a line crosses this parabola, the (x, y) points are the solutions of the system.

Graphically, the solutions are exactly the points where the curves intersect.

Quick check

Check your understanding with a question from this topic:

What is the positive solution to the equation x² - 7x + 10 = 0?

Your answer:

Enter a whole number, fraction (e.g. 3/4), or decimal (e.g. .75).

Worked examples

Example 1

What are the solutions to x² - 12x + 32 = 0?

Example 2

If x² - 8x + 7 = 0, what is the sum of the two solutions?

Example 3

The system of equations below has how many real solutions (x, y)?

y = x² + 2 y = 6x - 7

Common pitfalls

Forgetting the ± when square-rooting

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.

Sign errors in factoring

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.

Not setting the equation to zero first

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.

Misreading 'the' solution

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 - 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.

Watch & learn

Curated Khan Academy walkthroughs on Nonlinear Equations in 1 Variable and Systems of Equations in 2 Variables. They're complementary to this lesson — watch one if a written explanation isn't clicking, or after to reinforce.

Tracks your progress across lessons.

Try it yourself

5 practice questions on Nonlinear Equations in 1 Variable and Systems of Equations in 2 Variables, drawn from the question bank. The tutor is one click away if you get stuck.

Where this lesson fits

Nonlinear Equations in 1 Variable and Systems of Equations in 2 Variables

You are here

→Nonlinear Functions →Equivalent Expressions →Systems of 2 Linear Equations in 2 Variables