Question

If r and s are constants and r² +rx + 12 is equivalent to (x+3)(x + 5), what is the value of r?

F.:3
H. 7
J. 12
K. Cannot be determined from the given information

Answer 1

`\large\boxed{\sf r = 7 }`

Explanation:

Correct question:- If r and s are constants and r² +rx + 12 is equivalent to (x+3)(x + s), what is the value of r?

Here we are given that , the expression (x+3)(x+s) is equal to r² + rx + 12 .

Firstly, expand the expression (x+3)(x+s) as ,

`\implies (x+3)(x+s) \\`

`\implies x(x+s)+3(x+s) \\`

`\implies x^2 + xs + 3x + 3s \\`

Take out x as common,

`\implies x^2 + (3+s)x + 3s \\`

Now according to the question,

`\implies x^2 + (3+s)x + 3s = r^2 + rx + 12\\`

On comparing the respective terms , we get,

`\implies r = 3 + s \\`

`\implies 3s = 12 \\`

Solve the second equation to find out the value of s , so that we can substitute that in equation 1 to find "r" .

`\implies 3s = 12 \\`

`\implies s =(12)/(3)=\boxed{4} \\`

Now substitute this value in equation (1) as ,

`\implies r = 3 + s \\`

`\implies r = 3 + 4 \\`

`\implies \underline{\underline{ \red{ r = 7 }}} \\`

and we are done!

Answer 2

H. 7

Explanation:

Given x² + rx + 12 is equivalent to (x + 3)(x + s), equate the two expressions and expand the right side of the equation:

`\begin{aligned}x^2+rx+12&=(x + 3)(x + s)\\ x^2+rx+12&=x^2 + sx + 3x + 3s\\x^2+rx+12&=x^2 + (s+3)x + 3s\end{aligned}`

To find the value of r, first find the value of s.

The constant term of the right-hand side must be equal to the constant term of the left-hand side. Therefore:

`\implies 3s = 12`

Solve for s by dividing both sides of the equation by 3:

`\implies s = 4`

Compare the coefficients of the terms in x:

`\implies r = s + 3`

Substitute the value of s into the equation and solve for r:

`\begin{aligned} \implies r &= s + 3\\&= 4 + 3\\&= 7\end{aligned}`

Therefore, the value of r is 7.