Algebra Class 10: Radical Equations and Substitutions

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Questions and Answers

What substitution is used in Problem 1 to simplify the equation?

u = 3.5x
u = x+3
u = √(x+3) (correct)
u = √(x-3)

The radical equation in Problem 2 has a solution.

False (B)

What is the critical issue discovered when solving Problem 2?

The original problem has no solution when u is negative.

In Problem 3, the equation is transformed after letting u = √x; the next step is to rewrite x as __________.

u² Signup and view all the answers

Match the following problems to their conclusions:

Problem 1 = No valid solution Problem 2 = No solution due to negative u Problem 3 = One solution is invalid Signup and view all the answers

What is the substituted variable in the equation $x^4 - 5x^2 + 6 = 0$?

u = x^2 (C) Signup and view all the answers

The solution set for the equation $x^4 - 3x^2 + 2 = 0$ is {1, -1, √2, -√2}.

True (A) Signup and view all the answers

What values of x are derived from the equation $x^4 - 5x^2 + 6 = 0$?

±√3, ±√2 Signup and view all the answers

The new equation after substituting $u = x^2$ into $x^4 - 5x^2 + 6 = 0$ is $u^2 - 5u + ______ = 0$.

6 Signup and view all the answers

Match the following equations with their respective solutions:

x^4 - 5x^2 + 6 = 0 = {√3, -√3, √2, -√2} x^4 - 3x^2 + 2 = 0 = {1, -1, √2, -√2} Signup and view all the answers

Which of the following describes a conditional inequality?

It is true only for specific values of the variable involved. (D) Signup and view all the answers

An absolute inequality is true for all permissible values of the variable involved.

True (A) Signup and view all the answers

What does a parenthesis ')' or '(' signify in interval notation?

values are not included Signup and view all the answers

In set notation, {x ∈ R | x > -2} means x belongs to the set of real numbers such that x is greater than _____ .

-2 Signup and view all the answers

Match the following inequalities with their descriptions:

3x + 6 > 0 = conditional inequality 3x² - 6 ≤ 0 = quadratic inequality x ≤ 36 = includes endpoint (-2, ∞) = excludes -2 Signup and view all the answers

What is the solution to the inequality 5x - x ≥ x + 9?

x ≤ 3 (D) Signup and view all the answers

The interval notation for the solution x ≥ 6 is (-∞, 6).

False (B) Signup and view all the answers

What is the set notation for the inequality x - 1 ≤ 3x - 9 ≤ 2x + 5?

{x ∈ R | 4 ≤ x ≤ 14} Signup and view all the answers

The interval notation for the inequality 5x - x ≥ x + 9 can be expressed as (-∞, _____].

3 Signup and view all the answers

Match the following problems with their solutions:

Problem 1 = x ≤ 3 Problem 2 = x ≥ 6 Problem 3 = 4 ≤ x ≤ 14 Signup and view all the answers

What are the critical values found when solving the inequality x² + 2x - 3 > 0?

-3, 1 (C) Signup and view all the answers

The interval (-3, 1) results in a true statement for the inequality x² + 2x - 3 > 0.

False (B) Signup and view all the answers

What is the solution to the inequality x² + 2x - 3 > 0?

x < -3 or x > 1 Signup and view all the answers

In the solution process, the inequality x² + 2x - 3 = 0 is factored as (x + )(x - ) = 0.

3, 1 Signup and view all the answers

Match the intervals with their results for the inequality x² + 2x - 3 > 0:

(-∞, -3) = TRUE (-3, 1) = FALSE (1, ∞) = TRUE Signup and view all the answers

What is the first step in solving the equation $x^4 - 5x^2 + 4 = 0$?

Substituting $u = x^2$ (B) Signup and view all the answers

The solutions for the quadratic equation $u^2 - 5u + 4 = 0$ are $u = 2$ and $u = 3$.

False (B) Signup and view all the answers

What is the final solution for $x$ when solving the equation $x^4 - 5x^2 + 4 = 0$?

$x = ext{±1, ±2}$ Signup and view all the answers

For the equation $x^3 - 2x^2 + x + 1 = 0$, the substitution used was $u = x + _____$.

$1/2$ Signup and view all the answers

Which method is primarily used to solve the new quadratic equation after substitution?

Factoring (D) Signup and view all the answers

The equation $x^3 + 3x^2 - 4 = 0$ involves a substitution technique that is unclear.

True (A) Signup and view all the answers

When solving $u^2 - 5u + 4 = 0$, the solutions for $u$ are _ and _ .

1, 4 Signup and view all the answers

Match the polynomial equations with their corresponding substitution:

$x^4 - 5x^2 + 4 = 0$ = $u = x^2 $x^3 - 2x^2 + x + 1 = 0$ = $u = x + 1/2 $x^3 + 3x^2 - 4 = 0$ = $u = x + ext{unknown} Signup and view all the answers

What is the critical value found when solving the inequality $rac{2}{3x - 5} le 0$?

5/3 (A) Signup and view all the answers

The solution to the inequality $rac{2x-5}{x-5} le 3$ includes the point x = 6.

False (B) Signup and view all the answers

What is the resulting inequality after rewriting $rac{2x-5}{x-5} le 3$?

\frac{(2x - 5) - 3(x-5)}{x - 5} le 0 Signup and view all the answers

The test interval used to check the inequality $rac{2}{3x - 5} le 0$ was __________.

(-∞, 5/3) Signup and view all the answers

Match the critical values to their respective intervals:

5 = (-∞, 5) 10 = (10, ∞) 5/3 = (-∞, 5/3) 0 = (-∞, 5) Signup and view all the answers

Which of the following intervals represents the solution to the inequality $x^2 - x + 6 geq 0$?

(- orall, -2] igcup [3, orall) (C) Signup and view all the answers

The roots of the quadratic equation $x^2 - x + 6 = 0$ are $x = -2$ and $x = 3$.

False (B) Signup and view all the answers

What test point shows the inequality $x^2 - x + 6 geq 0$ is true for the interval $(- orall, -2)$?

x = -3 Signup and view all the answers

To solve the inequality $x^2 - x + 6 geq 0$, the first step is to __________ the quadratic.

factor Signup and view all the answers

Match each test point with its corresponding evaluation result for the inequality $x^2 - x + 6 geq 0$:

x = -3 = True x = 0 = False x = 4 = True Signup and view all the answers

What is the first critical value identified in the problem?

5 (B) Signup and view all the answers

The intervals of interest exclude the values where the expression is zero.

False (B) Signup and view all the answers

What is the interval notation for all values less than or equal to 5?

(-∞, 5] Signup and view all the answers

The second critical value identified is __________.

10 Signup and view all the answers

Match the intervals with their corresponding expression state:

(-∞, 5] = Expression is negative or zero [5, 10] = Expression is zero [10, ∞) = Expression is positive Signup and view all the answers

Flashcards

Quadratic in Form

An equation that can be rewritten as a standard quadratic equation using a substitution.

Substitution Strategy

A technique for solving equations that isn't immediately solvable. A variable is replaced with a simpler variable.