https://file.aiquickdraw.com/mathbot/file/17215940306393815t1ie.png

sOLVE 49,50 BY QUADRATIC FORM METHOD AND 64,65 BY ANY METHOD

Let's solve the specified problems using the given methods.

### Problem 49: \( x^{-4} - 9x^{-2} + 20 = 0 \)

First, we will make a substitution to convert this into a quadratic equation:
Let \( y = x^{-2} \). Thus, \( y^2 = x^{-4} \).

The equation becomes:
\[ y^2 - 9y + 20 = 0 \]

This is a quadratic equation. We can solve it using the quadratic formula:
\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = -9 \), and \( c = 20 \).

1. Calculate the discriminant:
\[ \Delta = (-9)^2 - 4(1)(20) = 81 - 80 = 1 \]

2. Find the roots:
\[ y = \frac{9 \pm \sqrt{1}}{2(1)} = \frac{9 \pm 1}{2} \]
\[ y_1 = \frac{10}{2} = 5 \]
\[ y_2 = \frac{8}{2} = 4 \]

Now, substitute back \( y = x^{-2} \):

For \( y_1 = 5 \):
\[ x^{-2} = 5 \]
\[ x^2 = \frac{1}{5} \]
\[ x = \pm \frac{1}{\sqrt{5}} = \pm \frac{\sqrt{5}}{5} \]

For \( y_2 = 4 \):
\[ x^{-2} = 4 \]
\[ x^2 = \frac{1}{4} \]
\[ x = \pm \frac{1}{2} \]

Thus, the solutions are:
\[ x = \pm \frac{\sqrt{5}}{5}, \pm \frac{1}{2} \]

### Problem 50: \(\frac{1}{x^4} - \frac{9}{x^2} + 8 = 0 \)

Let's use a similar substitution:
Let \( y = x^{-2} \). Thus, \( y^2 = x^{-4} \).

The equation becomes:
\[ y^2 - 9y + 8 = 0 \]

This is a quadratic equation. We can solve it using the quadratic formula:
\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = -9 \), and \( c = 8 \).

1. Calculate the discriminant:
\[ \Delta = (-9)^2 - 4(1)(8) = 81 - 32 = 49 \]

2. Find the roots:
\[ y = \frac{9 \pm \sqrt{49}}{2(1)} = \frac{9 \pm 7}{2} \]
\[ y_1 = \frac{16}{2} = 8 \]
\[ y_2 = \frac{2}{2} = 1 \]

Now, substitute back \( y = x^{-2} \):

For \( y_1 = 8 \):
\[ x^{-2} = 8 \]
\[ x^2 = \frac{1}{8} \]
\[ x = \pm \frac{1}{\sqrt{8}} = \pm \frac{\sqrt{8}}{8} = \pm \frac{\sqrt{2}}{2} \]

For \( y_2 = 1 \):
\[ x^{-2} = 1 \]
\[ x^2 = 1 \]
\[ x = \pm 1 \]

Thus, the solutions are:
\[ x = \pm \frac{\sqrt{2}}{2}, \pm 1 \]

### Problem 64: \( \frac{1}{2t} + \frac{2}{3t} = 2 \)

To solve this equation, we first find a common denominator for the fractions on the left side:
\[ \frac{1}{2t} + \frac{2}{3t} = \frac{3}{6t} + \frac{4}{6t} = \frac{7}{6t} = 2 \]

Now, solve for \( t \):
\[ \frac{7}{6t} = 2 \]
\[ 7 = 12t \]
\[ t = \frac{7}{12} \]

Thus, the solution is:
\[ t = \frac{7}{12} \]

### Problem 65: \( 3(2x + 1) = 4(x - 3) + 2 \)

Expand and simplify the equation:
\[ 6x + 3 = 4x - 12 + 2 \]
\[ 6x + 3 = 4x - 10 \]

Move all terms involving \( x \) to one side and constant terms to the other:
\[ 6x - 4x = -10 - 3 \]
\[ 2x = -13 \]
\[ x = -\frac{13}{2} \]

Thus, the solution is:
\[ x = -\frac{13}{2} \]

Would you like further details or explanations on any of these solutions?

Here are 5 questions you might want to ask next:
1. Can you solve another problem using the quadratic formula?
2. How do you solve a quadratic equation by factoring?
3. What are the steps to solving a rational equation?
4. Can you explain the substitution method in solving quadratic-form equations?
5. How do you solve equations with fractions and variables?

**Tip:** When solving equations with fractions, it's often helpful to multiply through by the least common denominator (LCD) to eliminate the fractions and simplify the equation.

Explore detailed solutions to quadratic and fractional equations including x^(-4) - 9x^(-2) + 20 = 0, 1/(x^4) - 9/(x^2) + 8 = 0, 1/(2t) + 2/(3t) = 2, and 3(2x + 1) = 4(x - 3) + 2. Learn step-by-step methods and formulas used to solve these equations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation