Solve Nonlinear Equations

Solving Nonlinear Equations

This pattern covers solving quadratic equations, absolute value equations, and rational equations. The SAT loves asking for a specific solution (the positive one, the negative one) or the sum of all solutions. Knowing shortcuts like Vieta's formulas can save significant time.

The Core Techniques

Quadratic equations: Get to $ax^2 + bx + c = 0$, then factor or use the quadratic formula $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Sum of solutions shortcut (Vieta's): For $ax^2 + bx + c = 0$, the sum of the two solutions is $-\dfrac{b}{a}$. You don't need to find each root individually.

Absolute value equations: $|E| = k$ splits into $E = k$ or $E = -k$ (when $k \geq 0$).

Rational equations: Multiply both sides by the denominator to clear the fraction, then solve the resulting polynomial.

Worked Examples

Example 1. $|y - 15| = 22$. What is the positive solution?

A) $7$
B) $22$
C) $37$
D) $-7$

Split: $y - 15 = 22$ → $y = 37$, or $y - 15 = -22$ → $y = -7$.
The positive solution is $37$.
Gotcha: Option D ($-7$) is the negative solution. Read carefully whether the question asks for the positive or negative one.
The answer is C.

Example 2. $|2x + 5| - 8 = 11$. What is the sum of the solutions?

First isolate the absolute value: $|2x + 5| = 19$.
Split: $2x + 5 = 19$ → $x = 7$, or $2x + 5 = -19$ → $x = -12$.
Sum: $7 + (-12) = -5$
Shortcut: For $|2x + 5| = k$, the two solutions are symmetric around $x = -\dfrac{5}{2}$. The sum is always $2 \times \left(-\dfrac{5}{2}\right) = -5$, regardless of $k$.
The answer is $-5$.

Example 3. $2k^2 = 12k - 3$. What is the sum of the solutions?

A) $12$
B) $\dfrac{3}{2}$
C) $-6$
D) $6$

Rearrange to standard form: $2k^2 - 12k + 3 = 0$.
Use Vieta's: sum $= -\dfrac{b}{a} = -\dfrac{-12}{2} = 6$.
Gotcha: Option A ($12$) forgets the negative sign in $-b/a$. Option B ($3/2$) gives $c/a$, which is the product of the roots, not the sum. Option C ($-6$) drops the negative: $-(-12)/2 = +6$, not $-6$.
The answer is D.

Example 4. $x - 4 = \dfrac{21}{x}$. What is the positive solution?

Multiply both sides by $x$: $x^2 - 4x = 21$
$x^2 - 4x - 21 = 0$
Factor: $(x - 7)(x + 3) = 0$, so $x = 7$ or $x = -3$.
The positive solution is $7$.
Gotcha: After clearing the fraction, don't forget you've introduced the possibility of $x = 0$ as extraneous. Here neither solution is zero, so both are valid. But always check that your solution doesn't make the original denominator zero.
The answer is $7$.

Example 5. $x^2 + 5x - 14 = 0$. What is a solution?

Factor: $(x + 7)(x - 2) = 0$, so $x = -7$ or $x = 2$.
If the question asks "what is a solution," either one works. Look at the answer choices and pick whichever appears.

Example 6. $(x - 3)^2 = 49$. What are the solutions?

Take the square root of both sides: $x - 3 = \pm 7$
$x = 3 + 7 = 10$ or $x = 3 - 7 = -4$
Gotcha: Don't forget the $\pm$. If you only write $x - 3 = 7$, you miss $x = -4$.

Example 7. $4x^2 - 12x + 9 = 0$. How many solutions does this have, and what is the sum?

Recognize this as $(2x - 3)^2 = 0$, so $x = \dfrac{3}{2}$ is the only solution (repeated root).
Sum of solutions: $\dfrac{3}{2}$ (just the one value).
You can verify: $-\dfrac{b}{a} = -\dfrac{-12}{4} = 3$... but wait, Vieta's counts the repeated root twice, giving sum $= 3$. For a single-answer SPR question, give the single solution $\dfrac{3}{2}$, but if asked for the "sum of all solutions" of a repeated root, it's still $\dfrac{3}{2}$ (one solution counted once).

What to Do on Test Day

• Sum of solutions shortcut: For $ax^2 + bx + c = 0$, the sum is $-\dfrac{b}{a}$. No need to find each root.
• Absolute value: Isolate $|E|$ first, then split into two cases. If $|E| = k$ with $k < 0$, there's no solution.
• Rational equations: Multiply by the denominator, solve, then check that no solution makes the denominator zero.
• "Positive solution" vs. "sum": Read carefully. These are different questions. One asks for a single root; the other asks for the total.
• Don't forget $\pm$. When you take a square root, you get two values. $(x-3)^2 = 49$ gives $x = 10$ and $x = -4$.
• Rearrange first. Get everything to one side in standard form ($= 0$) before factoring or applying formulas. A common mistake is applying Vieta's to an equation that isn't in standard form.