To solve: The proportion x2=3x+1

Question

To solve:  The proportion x2=3x+1

Vivian Soares · Accepted Answer

Approach.
The quotient of two numbers is called a ratio.
Ex:

\frac{5}{8} or 5 : 8

An equation indicating that two ratios are equal is called a proportion.
Ex:

\frac{3}{4} = \frac{15}{20}, \frac{p}{q} = \frac{r}{s}

In the proportion

\frac{p}{q} = \frac{r}{s}

, the numbers p and s are called extremes, and the numbers r and q are called the means.
In any proportion, the product of the extremes is equal to the product of the means.
Ex:

p s = r q

Two variables are said to be directly proportional or varies directly, if their ratio is a constant.
Ex:

\frac{a}{b} = k, a = k b

where k is a constant.
Two variables are said to be inversely proportional or varies inversely, if their product is a constant.
Ex:

a b = k, a = \frac{k}{b}

where k is a constant.
Variable a is said to be jointly proportional with b and c means

a = k b c

where k is a constant.
Given:

\frac{x}{2} = \frac{3}{x + 1}

Calculation:
Consider,

\frac{x}{2} = \frac{3}{x + 1}

By property of proportions, the product of the extremes is equal to the product of the means.

x \times (x + 1) = 2 \times 3

Apply distributive property

x^{2} + x = 6

Subtract 6 from both the sides

x^{2} + x - 6 = 6 - 6

x^{2} + x - 6 = 0

Factor the trinomial

x^{2} + 3 x - 2 x - 6 = 0

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

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

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

x = - 3 or x = 2

Therefore, the solution for the proportion

\frac{x}{2} = \frac{3}{x + 1} i s x = - 3 or x = 2

Final statement:
The solution for the proportion

\frac{x}{2} = \frac{3}{x + 1} i s x = - 3 or x = 2

Paineow · Accepted Answer

Step 1  Given: x2+3x+1  To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2 and x+1 is 2(x+1) Multiply x2 times x+1x+1. Multiply 3x+1 times 22  x(x+1)2(x+1)+3×22(x+1)  Since x(x+1)2(x+1) and 3×22(x+1) have the same denominator, add them by adding their numerators.  x(x+1)+3×22(x+1)  Do the multiplications in x(x+1)+3×2  x2+x+62(x+1)  Expand 2(x+1)  x2+x+62x+2

To solve: The proportion \frac{x}{2}=\frac{3}{x+1}

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