Solving Linear-Quadratic Systems

Where Does the Line Meet the Curve?

In this lesson:

• Predict 0, 1, or 2 intersections using the discriminant
• Solve by substitution; verify both solutions

What You Will Be Able to Do

By the end of this lesson, you should be able to:

1. Explain why this system type has 0, 1, or 2 solutions
2. Solve by substitution and back-substitute
3. Solve graphically; identify intersection points
4. Interpret: no intersection, tangent, or two points
5. Verify solutions in both original equations

Recall: Discriminant Predicts Quadratic Solutions

• : from
• : two real roots — : one root — : no real roots

If discriminant , what does the parabola look like?

How Many Times Can a Line Cross?

Line meets parabola in 2 points, 1 point (tangent), or 0 points

The Discriminant Predicts the Case

Substitution produces . Then:

Check discriminant first — it's a free preview.

The Substitution Method: Five Steps

1. Isolate in the linear equation
2. Substitute that expression into the quadratic
3. Simplify to
4. Solve the quadratic
5. Back-substitute each into linear for ; verify

Always: linear expression into quadratic equation.

Worked Example: Line Meets Parabola

Solve: and

Substitute for :

or

Back-substitute: and

Solutions: and . Verify in both equations. ✓

When the Line Is Tangent to the Parabola

Solve: and

, .  One solution:

Double root → tangent at .

Check-In: Predict the Number of Solutions

For and :

1. Substitute to get a quadratic in standard form
2. Compute
3. How many real solutions does this system have?

Try this before seeing the answer.

Circle Equations Are Quadratic Systems

: circle at origin, radius — quadratic in both variables.

CCSS example: Find intersections of and .

Estimate from the graph, then solve for exact values.

Solving the CCSS Circle Example

Substitute :

Back-Substitute Then Verify Both Equations

From : for each -value, find

Verify: ✓

When the Line Misses the Curve

System: and

Substitute: . Discriminant: .

No real solutions. The line is entirely above the parabola.

Negative discriminant → stop. No intersection points.

Common Error: Wrong Substitution Direction

Substituting quadratic into linear:

substituted into ... creates more complexity.

Correct direction: substitute into directly.

Linear expression into quadratic — every time.

Discriminant Detective: Predict Then Solve

Compute , predict intersections, then solve if real:

1. and
2. and
3. and
4. and

Guided Solve: Step by Step

Solve: and

Step 1: Substitute → ___

Step 2: Standard form → ___

Step 3: $x = $ ___

Step 4: $y = $ ___ for each .

Application: Rocket Meets a Sensor

A rocket's height:   ( = seconds, = feet)

A sensor triggers at height .

Set up the system:

or

The rocket is at height 21 twice — going up at 3s, coming down at 7s.

Your Turn: Line and Circle System

Solve algebraically: and

No hints. Show all steps, verify both solutions, interpret geometrically.

What You Can Do Now

✓ Discriminant predicts number of intersections before solving

✓ Substitute linear into quadratic; solve; back-substitute

✓ Verify in both original equations; interpret in context

Substitute linear into quadratic — not vice versa

Verify in BOTH equations — back-sub alone isn't enough

Coming Up: Graphs as Solution Sets

Next: HSA.REI.D.10-11 — the graph of an equation is the set of all (x, y) solutions.

Linear-quadratic systems show this connection: the solutions are exactly where the two graphs cross.

REI.D.11 generalizes: solutions to any system are the intersection points of the corresponding graphs.