(CNN) -

First came relief for Keegan Bradley, Hunter Mahan and Webb Simpson. Then thoughts turned to redemption.

The trio have been selected as wildcard picks by United States Ryder Cup captain Tom Watson for the biennial team clash with Europe that starts later in September.

And all three have memories from their previous appearance in golf's most enduring team contest they are desperate to erase.

For Bradley and Simpson -- both major winners -- it is banishing the demons from the "Miracle of Medinah" two years ago.

The USA was on course for its first Ryder Cup victory since 2008, before Europe overturned a 10-6 deficit on a dramatic final day.

Both Simpson and Bradley succumbed in the singles, to Ian Poulter and Rory McIlroy respectively, and are keen to make amends at Gleneagles in Scotland.

"I've made no secret of how important this team is to me and how badly I want to go back and win the Ryder Cup," Bradley told a press conference in New York at which Watson revealed his selections.

"I think this is a redemption year for a lot of guys who were on the team in 2012."

Bradley returned three points in Chicago thanks to a formidable partnership with five-time major champion Phil Mickelson.

The 2011 U.S. PGA Championship winner has had an up and down year but has recorded top 20 finishes in his last five events.

"He has the ability to hit the ball a long way and he came up to Gleneagles with me before the (British) Open Championship to play a practice round," Watson explained of his decision to pick the 28-year-old.

"He teamed up great with Phil Mickelson at Medinah and the most important thing he brings to it is his unbridled passion."

Simpson, the 2012 U.S. Open champion, was 15th on the qualification points list, lower than Chris Kirk who won the most recent PGA Tour event at the Deutsche Bank Championship.

He said it was "the greatest call" he'd ever received when Watson contacted him to tell him the news.

"It was a week that I will never forget," he said of 2012. "It was one of those weeks where it felt like we were going to win and it was kind of a shock Sunday afternoon.

"I'll never forget the feeling that I had watching the other team celebrate. They were all smiles and having fun, we had our heads down and it wasn't a good feeling.

"Two years has gone by very quickly and I remember all my matches like they were yesterday. I am eager to get back in the mix and try to get back the Ryder Cup."

Medinah was Simpson's first Ryder Cup appearance, and he won two matches and lost two. Those wins were emphatic -- 5&4 -- both alongside two-time Masters champion Bubba Watson.

And it was those big victories that persuaded captain Watson to select him.

"After struggling with a couple of different players, who was going to be the best for the team, I had a revelation this morning," the 64-year-old said.

"I looked at the last time the Ryder Cup was played and I see Webb Simpson 5&4, Webb Simpson 5&4 and I said 'that's got to be the guy'. Webb played in 2012, did very well.

"He has the ability to go real low but more important in our conversations, I liked his desire to be on the team. He showed such a strong desire to be in the team and he has had the performance this year to be one of the picks."

Watson's final pick -- Hunter Mahan -- is also out for redemption, but for the 2010 Ryder Cup, having missed the last transatlantic tussle.

The 2010 installment, at Celtic Manor in Wales, saw the outcome go down to Mahan's singles match with Graeme McDowell, the Northern Irishman defeating him to seal Europe's victory.

Mahan was a long way down the points list but has enjoyed a recent surge in form, finishing seventh at the U.S. PGA Championship before winning the recent Barclays tournament in the season-ending FedEx Playoffs.

"Losing lingers," Mahan said of his 2010 experience. "It's been four years, but it feels like yesterday. I remember walking off the green and everyone was having a big party, and I felt like I was walking by myself for 600 yards to the clubhouse.

"The guys had a tough time at Medinah and I had a tough time in 2010, so we have a lot to play for on this team. Winning is everything in golf and I could not have won a bigger tournament at the right time."

Earlier on Tuesday, European captain Paul McGinley chose Steven Gallacher, Ian Poulter and Lee Westwood as his three picks. The action begins on September 26.