SAT — mathHard5 min read

Linear-Quadratic Systems

Solve systems of one linear and one quadratic equation for the Digital SAT. Find intersection points between lines and parabolas.

Tutor AI Team2026-02-25T01:23:11.204Z

A linear-quadratic system consists of one linear equation and one quadratic equation. Graphically, this is a line intersecting a parabola. Such a system can have 0, 1, or 2 solutions depending on the relative positions of the line and parabola. The Digital SAT tests this by asking you to solve the system, find the number of solutions, or determine a parameter value.

Core Concepts

Three Cases

Case	Graphically	Solutions
Line intersects parabola twice	2 intersection points	2 solutions
Line is tangent to parabola	1 intersection point	1 solution
Line misses parabola	No intersection	0 solutions

Solving by Substitution

The linear equation is usually already solved for $y$ .
Substitute into the quadratic equation.
Solve the resulting quadratic (by factoring or quadratic formula).
Find the corresponding $y$ -values.

Example: $y = x + 1$ and $y = x^2 - 2x + 3$

Set equal: $x + 1 = x^2 - 2x + 3$

$0 = x^2 - 3x + 2 = (x - 1)(x - 2)$

$x = 1$ → $y = 2$ . $x = 2$ → $y = 3$ .

Solutions: $(1, 2)$ and $(2, 3)$ .

Using the Discriminant

After setting the equations equal, you get a quadratic. Apply the discriminant to determine the number of solutions without actually solving:

$\Delta > 0$ : 2 intersection points
$\Delta = 0$ : 1 intersection point (tangent)
$\Delta < 0$ : 0 intersection points

Finding Parameters

Example: For what value of $k$ is $y = x + k$ tangent to $y = x^2$ ?

$x + k = x^2$ → $x^2 - x - k = 0$

For tangency: $\Delta = 1 + 4k = 0$ → $k = -\frac{1}{4}$

Strategy Tips

Tip 1: Substitute to Eliminate $y$

Always substitute the linear equation into the quadratic. This gives a single-variable quadratic.

Tip 2: Use the Discriminant for "How Many Solutions"

If the question only asks for the number of solutions, don't solve — just compute the discriminant.

Tip 3: Remember to Find Both Coordinates

After finding $x$ -values, plug back to get $y$ -values. The solution is an ordered pair.

Tip 4: Graph on Desmos

Quickly visualise the intersection by graphing both equations.

Tip 5: If $y$ Isn't Isolated

If neither equation is solved for a variable, solve the simpler one (usually the linear equation) first.

Worked Example: Example 1

Problem

Solve $y = 2x$ and $y = x^2 - 3$ .

$2x = x^2 - 3$ → $x^2 - 2x - 3 = 0$ → $(x-3)(x+1) = 0$

$x = 3, y = 6$ and $x = -1, y = -2$

Solution

Worked Example: Example 2

Problem

How many solutions? $y = x + 5$ and $y = x^2 + 2x + 3$

$x + 5 = x^2 + 2x + 3$ → $x^2 + x - 2 = 0$

$\Delta = 1 + 8 = 9 > 0$ → 2 solutions

Solution

Worked Example: SAT-Style

Problem

The line $y = kx$ is tangent to the parabola $y = x^2 + 1$ . What is $k$ ?

$kx = x^2 + 1$ → $x^2 - kx + 1 = 0$

$\Delta = k^2 - 4 = 0$ → $k = \pm 2$

Solution

Worked Example: Example 4

Problem

Solve $y = -x + 4$ and $y = x^2 - 4x + 6$ .

$-x + 4 = x^2 - 4x + 6$ → $x^2 - 3x + 2 = 0$ → $(x-1)(x-2) = 0$

$x = 1, y = 3$ and $x = 2, y = 2$

Solutions: $(1, 3)$ and $(2, 2)$ .

Solution

Worked Example: Example 5

Problem

Does $y = 3$ intersect $y = x^2 + 4$ ?

$3 = x^2 + 4$ → $x^2 = -1$ — impossible over reals.

No intersection.

Solution

Practice Problems

Problem 1

Solve $y = x + 2$ and $y = x^2$ .

Problem 2

How many solutions? $y = 3x - 1$ and $y = x^2 + x + 1$ .

Problem 3

For what value of $c$ does $y = 2x + c$ intersect $y = x^2$ at exactly one point?

Problem 4

Solve $y = -2x + 1$ and $y = x^2 + 3x - 5$ .

Problem 5

A line $y = mx$ passes through two points on $y = x^2 - 4$ . Find the points when $m = 3$ .

Problem 6

Does the line $y = 5$ intersect the parabola $y = -x^2 + 4$ ?

Want to check your answers and get step-by-step solutions?

Common Mistakes

Solving the wrong equation. Always substitute the linear into the quadratic (or set them equal).
Forgetting to find $y$ -values. The solutions are points $(x, y)$ , not just $x$ -values.
Misidentifying the number of solutions. Use the discriminant of the resulting quadratic, not the original equations.
Errors in rearranging. When setting $mx + b = ax^2 + bx + c$ equal, be careful moving all terms to one side.
Not checking solutions. Plug back into both original equations.

Frequently Asked Questions

Can a line intersect a parabola more than twice?

No. A line can intersect a parabola at most 2 times.

What if the system has a quadratic and another quadratic?

The SAT primarily tests linear-quadratic systems. Quadratic-quadratic systems are rare.

How is tangency related to the discriminant?

A line is tangent to a parabola when the resulting quadratic has $\Delta = 0$ (one repeated solution).

Can I graph this on Desmos?

Absolutely — graph both equations and visually see the intersection points.

Is this under Algebra or Advanced Math on the SAT?

Advanced Math. It's one of the harder topics in the section.

Key Takeaways

✓
Substitute the linear equation into the quadratic to get a single-variable quadratic.
✓
Use the discriminant to find the number of solutions without solving.
✓
A line can intersect a parabola at 0, 1, or 2 points.
✓
$\Delta = 0$ means tangent (exactly one solution).
✓
Always find both coordinates of each solution.
✓
For parameter questions, set $\Delta = 0$ for tangency conditions.

Tags:

#sat#math#advanced-math#systems#quadratic

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Core Concepts

Three Cases

Solving by Substitution

Using the Discriminant

Finding Parameters

Strategy Tips

Tip 1: Substitute to Eliminate yyy

Tip 2: Use the Discriminant for "How Many Solutions"

Tip 3: Remember to Find Both Coordinates

Tip 4: Graph on Desmos

Tip 5: If yyy Isn't Isolated

Worked Example: Example 1

Worked Example: Example 2

Worked Example: SAT-Style

Worked Example: Example 4

Worked Example: Example 5

Practice Problems

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Common Mistakes

Frequently Asked Questions

Key Takeaways

Related Topics

Parallel Structure

Verb Tense and Consistency

Modifier Placement and Dangling Modifiers

Ready to Ace Your SAT math?

Tip 1: Substitute to Eliminate $y$

Tip 5: If $y$ Isn't Isolated