Virginia SOL Math / Algebra II / Equations & Inequalities

A2.EI.3

Systems Containing a Quadratic

What you'll learn

  • Solve a linear-quadratic system by substitution
  • Interpret the number of solutions graphically
  • Verify solutions in both equations

Words to know

System —
two or more equations solved together
Substitution —
replace one variable using another equation
Point of intersection —
a solution shared by both graphs
Linear-quadratic system —
a line paired with a parabola
Verify —
check a solution in every equation

Why it matters Many real situations pair a straight-line relationship with a curved one, and their intersections are the values that satisfy both at once.

Worked examples

Model ItI do

Solve the system y = x^2 and y = x + 2.

Substitute: x^2 = x + 2, so x^2 - x - 2 = 0, which factors as (x - 2)(x + 1) = 0. Then x = 2 or x = -1. Find y for each: (2, 4) and (-1, 1). These are the two points where the line meets the parabola.

Try It TogetherWe do

Solve y = x^2 and y = 9.

Substitute: x^2 = 9, so x = 3 or x = -3. The solutions are (3, 9) and (-3, 9).

Now YouYou do

Solve y = x^2 - 1 and y = 3.

Set the expressions equal and solve for x, then find y. Check your answer in the back of this chapter.

Practice

Mastered 0 of 11

Warm-up

Ease in with the basics.

  1. 1.Solve the system y = x^2 and y = 16.

  2. 2.How many solutions can a line and a parabola have?

  3. 3.Solve the system y = x^2 and y = 3x.

On Level

The core Quick Check from the book.

  1. 1.Solve y = x^2 and y = 4.

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

  3. 3.How many solutions can a line and a parabola have?

  4. 4.Is (1, 1) a solution of y = x^2?

  5. 5.Solve y = x^2 + 1 and y = 5.

Challenge

Stretch your thinking.

  1. 1.Solve the system y = x^2 - 2x + 1 and y = x + 1.

  2. 2.Solve the system y = x^2 + 1 and y = -2x - 2.

  3. 3.Solve the system y = x^2 - 4 and y = 2x - 1.