1) You need to use ajax.
2) For every button you can use a form such as:
<form method="post" action="approved.php" target="_self">
<td>
<input type="submit" name="submit" value="Approved">
<input type="hidden" name="quoteID" value="<?php echo $row["quote_id']?>'>
</td>
</form>
approved.php:
mysqli_connect
$approvedElement = $_POST['quoteID'];
$query = 'UPDATE ... WHERE `Quote ID` = \''.$quoteID.'\' ';
mysqli_query($query);
So before ajax I suggest you to learn basics about GET and POST methods.
In this example you need:
• form, to redirect the user to another page (approved.php, rejected.php)
• $_POST, in the second page to retrieve the ID of the approved element, and use it in the next step
• mysql_query, after you have correctly coded the query and successfully connected to the DB
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