The expression x2 + 10x + 18 can be written as (x + p)2 + q, where p and q are integers. Find the values of p…

Question

The expression x2 + 10x + 18 can be written as (x + p)2 + q, where p and q are integers. Find the values of p and q.

ScanSolve AI Tutor · Accepted Answer

Here are the solutions to questions 49 and 50.

49. The expression $x^2 + 10x + 18$ can be written as $(x+p)^2 + q$ . Find the values of $p$ and $q$ .

Expand the given form $(x+p)^2 + q$ .
$(x+p)^2 + q = x^2 + 2px + p^2 + q$
Compare the expanded form with the given expression $x^2 + 10x + 18$ .
By comparing the coefficients of $x$ :
$2p = 10$ $p = \frac{10}{2}$ $p = 5$
Compare the constant terms.
$p^2 + q = 18$
Substitute the value of $p=5$ into this equation:
$(5)^2 + q = 18$ $25 + q = 18$ $q = 18 - 25$ $q = -7$
The values are $p=5$ and $q=-7$ .
This corresponds to option C.

The final answer is $\boxed{\text{C. 5 and -7}}$ .

50. Solve: $2x - 3 \leq 4x + 5 < 3x + 7$ .

This is a compound inequality that can be split into two separate inequalities.
Solve the first inequality
$2x - 3 \leq 4x + 5$ .
Subtract $2x$ from both sides:
$-3 \leq 2x + 5$
Subtract 5 from both sides:
$-3 - 5 \leq 2x$ $-8 \leq 2x$
Divide by 2:
$\frac{-8}{2} \leq x$ $-4 \leq x$
Solve the second inequality
$4x + 5 < 3x + 7$ .
Subtract $3x$ from both sides:
$x + 5 < 7$
Subtract 5 from both sides:
$x < 7 - 5$ $x < 2$
Combine the solutions from Step 1 and Step 2.
We have $-4 \leq x$ and $x < 2$ .
Combining these gives:
$-4 \leq x < 2$
This corresponds to option B.

The final answer is $\boxed{\text{B. -4 \leq x < 2}}$ .

The expression x2 + 10x + 18 can be written as (x + p)2 + q, where p and q are integers. Find the values of p and q.