Complete lesson: quadratic equations, radical equations, and systems with nonlinear terms. Rules, strategies, solved examples, and 10 practice questions.
📖 What Are Nonlinear Equations?
A nonlinear equation is an equation where the variable appears with an exponent other than 1, inside a root, or in a more complex relationship. Unlike linear equations, nonlinear equations can have multiple solutions.
Quadratic Equation
\(x^2 - 5x + 6 = 0\)
Highest exponent is 2. Can have 0, 1, or 2 real solutions.
Radical Equation
\(\sqrt{x + 3} = 5\)
Variable inside a root. Must check for extraneous solutions.
Nonlinear System
\(y = x^2\) and \(y = x + 2\)
At least one equation is nonlinear. Can have multiple intersection points.
1. Quadratic Equations
\( ax^2 + bx + c = 0 \)
📌 RULE 1: Solving by Factoring
Step 1: Write the equation in the form \(ax^2 + bx + c = 0\). Step 2: Factor the quadratic expression. Step 3: Set each factor equal to zero and solve for \(x\).
If factoring is difficult, use the Quadratic Formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
The discriminant \(\Delta = b^2 - 4ac\) tells you the number of solutions:
\[
\Delta > 0 \rightarrow \text{2 real solutions}
\]
\[
\Delta = 0 \rightarrow \text{1 real solution}
\]
\[
\Delta < 0 \rightarrow \text{No real solutions}
\]
💡 Strategy — Quadratic Equations
1. Try factoring first (quickest).
2. If factoring doesn't work, use the Quadratic Formula.
3. Always check the discriminant to know how many solutions to expect.
📝 SOLVED EXAMPLE 1 — Quadratic (Factoring)
Solve: \(x^2 - 7x + 12 = 0\)
Step 1: \(x^2 - 7x + 12 = 0\)
Step 2: Factor: \((x - 3)(x - 4) = 0\)
Step 3: Set each factor to zero: \(x - 3 = 0\) or \(x - 4 = 0\)
Step 4: \(\color{var(--math)}{x = 3}\) or \(\color{var(--math)}{x = 4}\)
✅ \(x = 3, 4\)
💡 Tip: Always check your answers by plugging them back in.
📝 SOLVED EXAMPLE 2 — Quadratic (Quadratic Formula)
💡 Tip: The discriminant is \(25 > 0\), so 2 real solutions.
2. Radical Equations
\( \sqrt{expression} = \text{constant} \)
📌 RULE: Solving Radical Equations
Step 1: Isolate the radical term on one side of the equation. Step 2: Square both sides of the equation to eliminate the radical. Step 3: Solve the resulting equation. Step 4:ALWAYS check for extraneous solutions by substituting back into the original equation.
When you square both sides of an equation, you may introduce extraneous solutions — solutions that don't satisfy the original equation. Always check your answers!
📝 SOLVED EXAMPLE 3 — Radical Equation
Solve: \(\sqrt{2x - 1} = 3\)
Step 1: Isolate the radical: \(\sqrt{2x - 1} = 3\)
💡 Tip: \(x = -2\) is extraneous because the square root is always non-negative!
3. Nonlinear Systems of Equations
📌 RULE: Solving Nonlinear Systems
Step 1: Use substitution — solve one equation for one variable and substitute into the other equation. Step 2: Solve the resulting nonlinear equation. Step 3: Substitute back to find the other variable. Step 4: The system can have multiple solutions — list all ordered pairs.
💡 Strategy — Nonlinear Systems
1. Use substitution (it's usually easier than elimination for nonlinear systems).
2. Solve the quadratic or radical equation that results.
3. Don't forget to find both \(x\) and \(y\) values for each solution.
4. Check your answers in both original equations.
📝 SOLVED EXAMPLE 5 — Nonlinear System
Solve the system: \(y = x^2\) and \(y = x + 2\)
Step 1: Substitute: \(x^2 = x + 2\)
Step 2: Rearrange: \(x^2 - x - 2 = 0\)
Step 3: Factor: \((x - 2)(x + 1) = 0\)
Step 4: \(x = 2\) or \(x = -1\)
Step 5: For \(x = 2\): \(y = 2^2 = 4\) and \(y = 2 + 2 = 4\) ✓
Step 6: For \(x = -1\): \(y = (-1)^2 = 1\) and \(y = -1 + 2 = 1\) ✓
✅ \((2, 4)\) and \((-1, 1)\)
💡 Tip: The parabola \(y = x^2\) intersects the line \(y = x + 2\) at two points.
📝 SOLVED EXAMPLE 6 — Nonlinear System (No Solution)
Solve the system: \(y = x^2 + 1\) and \(y = -3\)
Step 1: Substitute: \(x^2 + 1 = -3\)
Step 2: Solve: \(x^2 = -4\)
Step 3: No real solution because \(x^2 \geq 0\)
✅ No real solution
💡 Tip: A parabola \(y = x^2 + 1\) never goes below \(y = 1\), so it never intersects \(y = -3\).
🧪 Practice Questions
Solve each problem using the rules above. Click "Show Answer" to see the full solution.
You've completed the Nonlinear Equations & Systems lesson. You now know how to solve quadratic equations, radical equations, and systems with nonlinear terms.