Problem 45 Solve each equation. $$ 9 x^... [FREE SOLUTION]

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Chapter 5: Problem 45

Solve each equation. $$ 9 x^{3}=16 x $$

Short Answer

Expert verified

x = 0, x = \frac{4}{3}, x = -\frac{4}{3}

Step by step solution

Set the equation to zero

Start by moving all terms to one side of the equation to set it to zero: \[9x^{3} - 16x = 0\]

Factor out the common term

Factor out the common term, which in this case is $x$: \[x(9x^{2} - 16) = 0\]

Solve for x using the Zero Product Property

The Zero Product Property states that if a product of factors equals zero, then at least one of the factors must be zero. Set each factor to zero and solve: 1. $x = 0$2. $9x^{2} - 16 = 0$

Solve the quadratic equation

Solve the quadratic equation $9x^{2} - 16 = 0$ by isolating $x$: \[9x^{2} = 16\]\[x^{2} = \frac{16}{9}\]\[x = \pm \sqrt{\frac{16}{9}}\]\[x = \pm \frac{4}{3}\]

Compile the solutions

Combine all the solutions from the steps: \[x = 0, x = \frac{4}{3}, x = -\frac{4}{3}\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Zero Product Property

When solving cubic equations, understanding the Zero Product Property is essential. The Zero Product Property states that if the product of two factors is zero, then at least one of the factors must be zero. This is written mathematically as: if $A * B = 0$, then either $A = 0$ or $B = 0$. In the context of our given equation 9x鲁 - 16x = 0, we factor it into x(9x虏 - 16) = 0.
Setting each factor to zero, we get:

x = 0
9x虏 - 16 = 0

By using the Zero Product Property, we can find the values of x that satisfy the equation.

Factoring Polynomials

Factoring polynomials is a crucial step in solving higher degree equations. In our example, the polynomial 9x鲁 - 16x is factored by taking out a common term, x, giving us x(9x虏 - 16) = 0. Factoring simplifies the polynomial, making it easier to solve.
Steps to factor a polynomial:

Identify the greatest common factor (GCF).
Factor out the GCF.
Simplify the remaining expression.

In the operation, x is the GCF, and after factoring it out, we get the polynomial in a simplified form, allowing us to apply the Zero Product Property.

Quadratic Equations

Quadratic equations have the form $ax虏 + bx + c = 0$. In our exercise, after factoring out x, we were left with the quadratic equation 9x虏 - 16 = 0.
To solve the quadratic equation, we isolate x as follows:

$9x虏 - 16 = 0$
Move constant term: $9x虏 = 16$
Divide by coefficient: $x虏 = \frac{16}{9}$
Simplify to find x: $x = \frac{4}{3}$ or $x = -\frac{4}{3}$

Quadratic equations can usually be solved by factoring, completing the square, or using the quadratic formula. In our case, since it was already in factorable form, we solved it directly.

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91影视

Solve each equation. $$ 9 x^{3}=16 x $$

Short Answer

Step by step solution

Set the equation to zero

Factor out the common term

Solve for x using the Zero Product Property

Solve the quadratic equation

Compile the solutions

Key Concepts

Zero Product Property

Factoring Polynomials

Quadratic Equations

One App. One Place for Learning.

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