Algebra II Exam Practice Problems
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Questions and Answers

What does infinitely many solutions mean?

infinitely many solutions

Solve the system: 4x + 3y = 12 and 2x + 5y = 6.

(3, 0)

If x = 482 adults and y = 273 students, what is the scenario about?

There were 755 people watching a Highland Park basketball game.

Find the numbers if x = 2, y = 5, z = -3 and the conditions are given.

<p>x = 2, y = 5, z = -3</p> Signup and view all the answers

Solve for x in | 3x + 4 | - 1 = 29.

<p>x = 2/3, -10/3</p> Signup and view all the answers

Solve for x in | x - 2 | = 2x - 10.

<p>x = 8</p> Signup and view all the answers

Solve for x in -3 | 4x - 3 | = -27.

<p>x = -3/2, 3</p> Signup and view all the answers

What is the solution set for the inequality | 2x - 3 | < 9?

<p>-3 &lt; x &lt; 6</p> Signup and view all the answers

What is the solution for the inequality | 3x | < -10?

<p>no solution</p> Signup and view all the answers

What is the solution set for - | 6x + 3 | < 15?

<p>-3 &lt; x &lt; 2</p> Signup and view all the answers

Find the solution for -2 | 2x - 4 | - 3 > -19.

<p>-2 &lt; x &lt; 6</p> Signup and view all the answers

What is the solution for 2 + 3(3 - x) < -(2x + 1)?

<p>x &gt; 12</p> Signup and view all the answers

What is the solution for 2/5 + 4/5x > 1/15?

<p>x &gt; -5/12</p> Signup and view all the answers

Solve the system by graphing: y + 3 < 2x and 2y < x + 2.

<p>Graphing the inequalities</p> Signup and view all the answers

What are the vertex coordinates for the parabola y = x^2 - 2x - 15?

<p>(-1/2, -25/4)</p> Signup and view all the answers

What is the y-intercept of the parabola y = x^2 - 2x - 15?

<p>(0, -6)</p> Signup and view all the answers

What are the roots of the parabola y = x^2 - 2x - 15?

<p>(-3, 0)(2, 0)</p> Signup and view all the answers

What direction does the parabola y = x^2 - 2x - 15 open?

<p>up</p> Signup and view all the answers

What are the vertex coordinates for the parabola y = x^2 - 2x - 15?

<p>(1, -16)</p> Signup and view all the answers

What are the roots of the parabola y = x^2 - 2x - 15?

<p>(5, 0)(-3, 0)</p> Signup and view all the answers

Solve the quadratic equation 4x^2 + 8x + 3 = 0.

<p>x = -1/2, -3/2</p> Signup and view all the answers

Solve the quadratic equation 8b^2 + 10b = 3.

<p>x = 1/4, -3/2</p> Signup and view all the answers

Solve for x in the equation 4x^2 + 32 = 0.

<p>x = +2i ext{ root } 2, -2i ext{ root } 2</p> Signup and view all the answers

Find the solutions for a^3 = 49a.

<p>a = 0, -7, 7</p> Signup and view all the answers

Solve the equation 6x^2 - 7x = 5.

<p>x = 5/3, -1/2</p> Signup and view all the answers

Solve the equation 18u^2 - 3u = 1.

<p>u = -1/6, 1/3</p> Signup and view all the answers

Simplify the expression (2 + 5i) - (4 - 6i).

<p>-2 + 11i</p> Signup and view all the answers

Simplify i^47.

<p>-i</p> Signup and view all the answers

Simplify the expression (4i - 2)(4i - 2).

<p>-12 - 16i</p> Signup and view all the answers

Solve by completing the square for x^2 - 4x - 21 = 0.

<p>x = -3, 7</p> Signup and view all the answers

Solve by completing the square for 4x^2 - 8x = -3.

<p>x = 3/2, 1/2</p> Signup and view all the answers

Solve by completing the square for x^2 - 2x + 2 = 0.

<p>x = 1+i, 1-i</p> Signup and view all the answers

Solve by completing the square for 4x^2 - 16x + 7 = 0.

<p>x = 7/2, 1/2</p> Signup and view all the answers

What is the vertex of the parabola y = -4x^2 + 16x - 11?

<p>(2, 5)</p> Signup and view all the answers

What is the axis of symmetry for the parabola y = -4x^2 + 16x - 11?

<p>x = 2</p> Signup and view all the answers

What is the width of the parabola y = -4x^2 + 16x - 11?

<p>narrow</p> Signup and view all the answers

What is the domain of the parabola y = -4x^2 + 16x - 11?

<p>(-inf, inf)</p> Signup and view all the answers

What is the range of the parabola y = -4x^2 + 16x - 11?

<p>(-inf, 5]</p> Signup and view all the answers

What is the vertex of the parabola y = -(x - 2)^2 + 5?

<p>(2, 5)</p> Signup and view all the answers

What is the axis of symmetry for the parabola y = -(x - 2)^2 + 5?

<p>x = 2</p> Signup and view all the answers

What is the width of the parabola y = -(x - 2)^2 + 5?

<p>same</p> Signup and view all the answers

What is the domain of the parabola y = -(x - 2)^2 + 5?

<p>(-inf, inf)</p> Signup and view all the answers

What is the range of the parabola y = -(x - 2)^2 + 5?

<p>(-inf, 5]</p> Signup and view all the answers

What is the vertex of the parabola y = 3x^2 - 9?

<p>(0, -9)</p> Signup and view all the answers

What is the axis of symmetry for the parabola y = 3x^2 - 9?

<p>x = 0</p> Signup and view all the answers

What is the width of the parabola y = 3x^2 - 9?

<p>narrow</p> Signup and view all the answers

What is the domain of the parabola y = 3x^2 - 9?

<p>(-inf, inf)</p> Signup and view all the answers

What is the range of the parabola y = 3x^2 - 9?

<p>[-9, inf)</p> Signup and view all the answers

Write the equation in standard form for the parabola that passes through the points (-2, 10), (-1, 3), (0, 10).

<p>y = 7x^2 + 14x + 10</p> Signup and view all the answers

Identify the vertex of the parabola with vertex at (6, -22).

<p>vertex: (6, -22)</p> Signup and view all the answers

What are the roots of the parabola with roots at (11.416, 0) and (.584, 0)?

<p>roots: (11.416, 0)(.584, 0)</p> Signup and view all the answers

What is the domain for the parabola with inequality domain stated as all real numbers?

<p>inequality domain: all real numbers</p> Signup and view all the answers

What is the range for the parabola with an inequality stated as y > -22?

<p>inequality range: y &gt; -22</p> Signup and view all the answers

What is the interval domain for the parabola?

<p>interval domain: (-inf, inf)</p> Signup and view all the answers

What is the interval range for the parabola?

<p>interval range: [-22, inf)</p> Signup and view all the answers

Study Notes

Infinitely Many Solutions

  • Refers to situations in systems of equations where multiple combinations of variable values satisfy all equations.

Solving Linear Equations

  • Solution for 4x + 3y = 12 and 2x + 5y = 6 is (3,0).
  • A Highland Park basketball game involved 482 adults and 273 students, with total attendance of 755 and ticket sales totaling $1614.50.

System of Equations with Variables

  • Three numbers that sum to 4: x = 2, y = 5, z = -3. The second number exceeds twice the first by 1, and the sum of the first and third equals -1.

Absolute Value Equations

  • Solving 5 | 3x+4 | -1 = 29 results in x = 2/3, -10/3.
  • For | x-2 | = 2x-10, the solution is x = 8.
  • For -3 | 4x-3 | = -27, solutions are x = -3/2, 3.

Inequalities and Graphing

  • | 2x-3 | < 9 yields -3 < x < 6; graph includes this interval.
  • | 3x | < -10 has no solution since absolute values cannot be negative.
  • For the inequality - | 6x+3 | < 15, the solution set is -3 < x < 2.

Further Inequalities

  • For -2 | 2x-4 | -3 > -19, the solution yields -2 < x < 6.
  • A solution set of 2 + 3(3-x) < - (2x+1) results in x being greater than 12.
  • Solve the inequality 2/5 + 4/5x > 1/15, yielding x > -5/12.

Graphing and Parabolas

  • Solve the system by graphing: y + 3 < 2x and 2y < x + 2.
  • Given the quadratic equation y = x^2 - 2x - 15, the vertex is (-1/2, -25/4), y-intercept is (0, -6), axis of symmetry is x = -1/2, roots are (-3,0) and (2,0), and it opens upwards.

Quadratic Formula Solutions

  • Quadratic equation 4x^2 + 8x + 3 = 0 solved yields x = -1/2 and -3/2.
  • For 8b^2 + 10b = 3, solutions are x = 1/4 and -3/2.
  • Solving 4x^2 + 32 = 0 results in complex roots: x = ±2i√2.

Polynomial and Complex Solutions

  • For a^3 = 49a, solutions are a = 0, -7, and 7.
  • 6x^2 - 7x = 5 yields solutions x = 5/3 and -1/2.
  • 18u^2 - 3u = 1 leads to u = -1/6 and 1/3.

Simplifying Complex Numbers

  • Simplification of (2 + 5i) - (4 - 6i) gives -2 + 11i.
  • i^47 simplifies to -i.
  • (4i - 2)(4i - 2) simplifies to -12 - 16i.

Completing the Square

  • Completing the square for x^2 - 4x - 21 = 0 yields solutions x = -3 and 7.
  • The equation 4x^2 - 8x = -3 results in solutions x = 3/2 and 1/2.
  • Completing the square for x^2 - 2x + 2 = 0 provides solutions x = 1+i and 1-i.
  • For 4x^2 - 16x + 7 = 0, solutions are x = 7/2 and 1/2.

Vertex Form and Characteristics of Parabolas

  • For y = -4x^2 + 16x -11, find vertex (2, 5), axis of symmetry x = 2, opens down, domain (-∞, ∞), and range (-∞, 5].
  • The second parabola y = -(x-2)^2 + 5 shares vertex (2, 5) and axis of symmetry x = 2, with the same characteristics.
  • For y = 3x^2 - 9, the vertex is (0, -9), opens up, domain (-∞, ∞), range [-9, ∞).

Standard Form for Parabolas

  • Transform the equation y = 7x^2 + 14x + 10 through given points to standard form.
  • Inequalities related to the parabola roots and domain/range: vertex at (6, -22) leads to inequality domain of all real numbers and range y > -22.

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This quiz features a series of flashcards designed to help you practice key concepts in Algebra II. You'll encounter problems related to solving equations and understanding the concept of infinitely many solutions. Perfect for exam preparation!

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