Question Statement

A welder is designing a triangular support structure for a building. The structure is made up of iron beams intersecting at the origin. The equations of the lines representing the beams are given by the joint equation $2 x^{3} - 8 x y^{2} = 0$ . Find the equations of the individual beams.

Background and Explanation

This problem involves factorizing a homogeneous cubic equation to find the equations of three straight lines passing through the origin. When multiple lines intersect at a point, their combined equation can be expressed as a product of their individual linear equations set equal to zero.

Solution

We are given the joint equation representing three beams (lines) passing through the origin:

$2 x^{3} - 8 x y^{2} = 0$

Step 1: Factor out the common term

First, we identify that both terms contain a common factor of $2 x$ :

$2 x (x^{2} - 4 y^{2}) = 0$

Step 2: Factor the quadratic expression

The expression inside the parentheses, $x^{2} - 4 y^{2}$ , is a difference of squares. Recall that $a^{2} - b^{2} = (a - b) (a + b)$ . Here, $a = x$ and $b = 2 y$ (since $4 y^{2} = (2 y)^{2}$ ):

$2 x (x - 2 y) (x + 2 y) = 0$

Step 3: Apply the zero product property

For the product of three factors to equal zero, at least one of the factors must be zero. This gives us three separate equations:

$2 x = 0, x - 2 y = 0, x + 2 y = 0$

Step 4: Simplify the equations

Simplifying the first equation by dividing by 2:

$x = 0, x - 2 y = 0, x + 2 y = 0$

Thus, the equations of the three beams are:

$x = 0$ (the y-axis)
$x - 2 y = 0$ or $x = 2 y$
$x + 2 y = 0$ or $x = - 2 y$

Key Formulas or Methods Used

Factorization of common terms: Extracting $2 x$ from the polynomial
Difference of squares formula: $a^{2} - b^{2} = (a - b) (a + b)$
Zero product property: If $A \cdot B \cdot C = 0$ , then $A = 0$ or $B = 0$ or $C = 0$
Homogeneous equations: Recognition that $2 x^{3} - 8 x y^{2} = 0$ represents lines through the origin

Summary of Steps

Factor out the greatest common factor $2 x$ from the given equation $2 x^{3} - 8 x y^{2} = 0$
Factor the remaining quadratic expression $x^{2} - 4 y^{2}$ using the difference of squares formula
Set each of the three factors equal to zero separately
Simplify the resulting equations to obtain the three line equations: $x = 0$ , $x - 2 y = 0$ , and $x + 2 y = 0$

Question Statement

Background and Explanation

Solution

We are given the joint equation representing three beams (lines) passing through the origin:

$2 x^{3} - 8 x y^{2} = 0$

Step 1: Factor out the common term

First, we identify that both terms contain a common factor of $2 x$ :

$2 x (x^{2} - 4 y^{2}) = 0$

Step 2: Factor the quadratic expression

The expression inside the parentheses, $x^{2} - 4 y^{2}$ , is a difference of squares. Recall that $a^{2} - b^{2} = (a - b) (a + b)$ . Here, $a = x$ and $b = 2 y$ (since $4 y^{2} = (2 y)^{2}$ ):

$2 x (x - 2 y) (x + 2 y) = 0$

Step 3: Apply the zero product property

For the product of three factors to equal zero, at least one of the factors must be zero. This gives us three separate equations:

$2 x = 0, x - 2 y = 0, x + 2 y = 0$

Step 4: Simplify the equations

Simplifying the first equation by dividing by 2:

$x = 0, x - 2 y = 0, x + 2 y = 0$

Thus, the equations of the three beams are:

$x = 0$ (the y-axis)
$x - 2 y = 0$ or $x = 2 y$
$x + 2 y = 0$ or $x = - 2 y$

Key Formulas or Methods Used

Factorization of common terms: Extracting $2 x$ from the polynomial
Difference of squares formula: $a^{2} - b^{2} = (a - b) (a + b)$
Zero product property: If $A \cdot B \cdot C = 0$ , then $A = 0$ or $B = 0$ or $C = 0$
Homogeneous equations: Recognition that $2 x^{3} - 8 x y^{2} = 0$ represents lines through the origin

Summary of Steps

Factor out the greatest common factor $2 x$ from the given equation $2 x^{3} - 8 x y^{2} = 0$
Factor the remaining quadratic expression $x^{2} - 4 y^{2}$ using the difference of squares formula
Set each of the three factors equal to zero separately
Simplify the resulting equations to obtain the three line equations: $x = 0$ , $x - 2 y = 0$ , and $x + 2 y = 0$

6.3 Q-9

Question Statement

Background and Explanation

Solution

Key Formulas or Methods Used

Summary of Steps

6.3 Q-9

Question Statement

Background and Explanation

Solution

Key Formulas or Methods Used

Summary of Steps