Learnig Task 34. Solve the following equations transformable to quadratic equations​

To transform rational algebraic expressions to quadratic equations, equate first everything to zero. This might mean changing the sign or operation of the last term after moving it to the left side of the equation. Remove the denominators using the least common denominator. Simplify the equations. Combine the similar terms when necessary.

1. 6x² + 22x + 12
2. x²  - 5x - 10 = 0
3. x² - 2x + 1 = 0

1. [tex]\frac{1}{x}[/tex] + [tex]\frac{x}{2}[/tex] = [tex]\frac{11}{6}[/tex]

Equate everything to zero.

[tex]\frac{1}{x}[/tex] + [tex]\frac{x}{2}[/tex] = [tex]\frac{11}{6}[/tex]

[tex]\frac{1}{x}[/tex] + [tex]\frac{x}{2}[/tex] - [tex]\frac{11}{6}[/tex] = 0

Using the least common denominator, eliminate the denominators of each terms.

(x)(2)(-6)[[tex]\frac{1}{x}[/tex] + [tex]\frac{x}{2}[/tex] - [tex]\frac{11}{6}[/tex] = 0]

(2)(-6)(1) + (x)(-6)(x) - (x)(2)(11) = 0

-12 + (-6x²) - 22x = 0

Write the equation in standard form.

-6x² - 22x - 12 = 0

Remove the negative coefficient of x by multiplying both sides of the equation by -1.

-1[-6x² - 22x - 12 = 0]

6x² + 22x + 12 = 0

2. [tex]\frac{2}{x}[/tex] - [tex]\frac{x - 1}{5}[/tex] = [tex]\frac{-4}{5}[/tex]

Equate everything to zero.

[tex]\frac{2}{x}[/tex] - [tex]\frac{x - 1}{5}[/tex] = [tex]\frac{-4}{5}[/tex]

[tex]\frac{2}{x}[/tex] - [tex]\frac{x - 1}{5}[/tex] + [tex]\frac{4}{5}[/tex] = 0

Using the least common denominator, eliminate the denominators of each terms.

5x [ [tex]\frac{2}{x}[/tex] - [tex]\frac{x - 1}{5}[/tex] + [tex]\frac{4}{5}[/tex] = 0]

(5)(2) - (x)(x - 1) + (x)(4) = 0

10 - x² + x + 4x = 0

Combine the similar terms.

10 - x² + 5x = 0

Write the equation in standard form.

-x² + 5x + 10 = 0

Remove the negative coefficient of x by multiplying both sides of the equation by -1.

-1[-x² + 5x + 10 = 0]

x²  - 5x - 10 = 0

3. [tex]\frac{2}{x - 3}[/tex] + [tex]\frac{x}{2}[/tex] = [tex]\frac{-1}{2}[/tex]

Equate everything to zero.

[tex]\frac{2}{x - 3}[/tex] + [tex]\frac{x}{2}[/tex] = [tex]\frac{-1}{2}[/tex]

[tex]\frac{2}{x - 3}[/tex] + [tex]\frac{x}{2}[/tex] + [tex]\frac{1}{2}[/tex] = 0

Using the least common denominator, eliminate the denominators of each terms.

(x - 3)(2)[[tex]\frac{2}{x - 3}[/tex] + [tex]\frac{x}{2}[/tex] + [tex]\frac{1}{2}[/tex] = 0]

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

4 + x² -3x + x - 3 = 0

Combine the similar terms.

x² - 2x + 1 = 0

Write the equation in standard form.

x² - 2x + 1 = 0

How to transform algebraic expressions to quadratic equations:

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