SAT Systems with nonlinear equations

Where a line and a curve meet — solve or count intersections.

4% of MathMath · Advanced Math3 question types

~2per test

How to score it

Substitute the linear equation into the quadratic, then solve.
“Intersect at exactly one point” means the quadratic has one root → discriminant = 0.
A line tangent to a parabola touches at exactly one point.

Common traps

Keeps an extraneous intersection.
“Exactly one point” not recognized as a discriminant condition.
Reads only one of two intersection points off a graph.

The 3 question types, with real examples

Line and quadratic intersect

“The graphs intersect at (x, y). What is the value of [y / x²]?”

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From the question bankMedium

$y = - x^{2} + 9$ $y = 2x + 6$ The system of equations above has solutions $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ . What is the value of $x_{1} + x_{2}$ ?

A $-2$ ✓
B $2$
C $12$
D $-1$

Why A

Substituting $y = 2x + 6$ into the first equation yields $2x + 6$ = - $x^{2}$ + 9. Rearranging gives $x^{2}$ + $2x - 3 = 0$ . By Vieta's formulas, the sum of the roots is -2/1 = -2.

Graph → which solution

“What is the solution (x, y) to this system?”

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From the question bankMedium

In the xy-plane, the graph of $y = x^{2} + 2x + 5$ intersects the graph of $y = 6x + 1$ at exactly one point.

What is the solution (x, y) to this system?

A $(1, 8)$
B $(2, 13)$ ✓
C $(-2, 13)$
D $(2, 9)$

Why B

$Setting x^{2} + 2x + 5 = 6x + 1 gives x^{2} - 4x + 4 = 0, which factors as (x - 2)^{2} = 0, so x = 2. Substituting into y = 6x + 1 gives y = 13.$

Identify possible value of x from the system

“[eq1] [eq2]. If (x, y) is a solution to the system, which of the following could be the value of x?”

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From the question bankMedium

$x^{2} + y^{2} = 25$ $y = x + 1$ The system of equations above has solutions $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ . What is the value of $y_{1} y_{2}$ ?

A $-12$ ✓
B $12$
C $-3$
D $-11$

Why A

Substituting $x + 1$ for y in the first equation yields $x^{2}$ + $(x + 1)^{2} = 25$ . Expanding gives $x^{2}$ + $x^{2}$ + $2x + 1 = 25$ , which simplifies to $2x^{2} + 2x - 24 = 0$ or $x^{2}$ + $x - 12 = 0$ . Since $y = x + 1$ , the product $y_{1} y_{2}$ = ( $x_{1}$ + 1)( $x_{2}$ + 1) = $x_{1} x_{2} + x_{1} + x_{2}$ + 1. By Vieta's formulas, $x_{1} x_{2}$ = -12 and $x_{1} + x_{2}$ = -1, so $y_{1} y_{2}$ = -12 + (-1) + 1 = -12.

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SAT Systems with nonlinear equations

How to score it

Common traps

The 3 question types, with real examples

Line and quadratic intersect

Graph → which solution

Identify possible value of x from the system

More Advanced Math skills

Knowing the pattern is step one. Drilling it is how you score.